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Fang, Pew-Thian Yap, Senior Member, IEEE, Weili Lin, Hongtu Zhu, and Mingxia Liu, Senior +Member, IEEE +Abstract—Unsupervised domain adaptation (UDA) via deep learning has attracted appealing attention for tackling domain-shift +problems caused by distribution discrepancy across different domains. Existing UDA approaches highly depend on the accessibility of +source domain data, which is usually limited in practical scenarios due to privacy protection, data storage and transmission cost, and +computation burden. To tackle this issue, many source-free unsupervised domain adaptation (SFUDA) methods have been proposed +recently, which perform knowledge transfer from a pre-trained source model to unlabeled target domain with source data inaccessible. +A comprehensive review of these works on SFUDA is of great significance. In this paper, we provide a timely and systematic literature +review of existing SFUDA approaches from a technical perspective. Specifically, we categorize current SFUDA studies into two groups, +i.e., white-box SFUDA and black-box SFUDA, and further divide them into finer subcategories based on different learning strategies +they use. We also investigate the challenges of methods in each subcategory, discuss the advantages/disadvantages of white-box and +black-box SFUDA methods, conclude the commonly used benchmark datasets, and summarize the popular techniques for improved +generalizability of models learned without using source data. We finally discuss several promising future directions in this field. +Index Terms—Domain adaptation, source-free, unsupervised learning, survey. +! +1 +INTRODUCTION +D +EEP learning, based on deep neural networks with rep- +resentation learning, has emerged as a promising tech- +nique and made remarkable progress over the past decade, +covering the field of computer vision [1], [2], medical data +analysis [3], [4], natural language processing [5], [6], etc. For +problems with multiple domains (e.g., different datasets or +imaging sites), the typical learning process of a deep neural +network is to transfer the model learned on a source domain +to a target domain. However, performance degradation is +often observed when there exists a distribution gap between +the source and target domains, which is termed “domain +shift” problem [7]–[9]. To tackle this problem, various do- +main adaptation algorithms [10], [11] have been proposed +to perform knowledge transfer by reducing inter-domain +distribution discrepancy. To avoid intensive burden of data +annotation, unsupervised domain adaptation has achieved +much progress [12]–[15]. As illustrated in Fig. 1 (a), unsuper- +vised domain adaptation aims to transfer knowledge from a +labeled source domain to a target domain without accessing +any target label information. +Existing deep learning studies on unsupervised domain +adaptation highly depend on the accessibility of source data, +which is usually limited in practical scenarios due to the +following possible reasons. (1) Data privacy protection. Many +source datasets containing confidential information, such as +medical and facial data, are not available to third parties +due to privacy and security protection. (2) Data storage and +• +Y. Fang, P.-T. Yap, W. Lin and M. Liu are with the Department +of Radiology and Biomedical Research Imaging Center, University of +North Carolina at Chapel Hill, Chapel Hill, NC 27599, United States. +H. Zhu is with the Department of Biostatistics, University of North +Carolina at Chapel Hill, NC 27599, USA. Corresponding author: M. Liu +(mxliu@med.unc.edu). +transmission cost. The storage and transmission of large- +scale source datasets, such as ImageNet [16], could bring +much economic burden. (3) Computation burden. Training on +extremely large source datasets requires high computational +resources, which is not practical, especially in real-time +deployment cases. Thus, there is a high demand for source- +free unsupervised domain adaptation (SFUDA) methods +that transfer a pre-trained source model to the unlabeled +target domain without accessing any source data [17]–[20]. +Many promising SFUDA algorithms have been devel- +oped recently to address problems in the fields of seman- +tic segmentation [21], image classification [22], object de- +tection [23], face anti-spoofing [24], etc. A comprehensive +review of current studies on SFUDA as well as an outlook +on future research directions are urgently needed. Liu et +al. [25] present a review on data-free knowledge transfer, +where SFUDA only accounts for part of the review and +the taxonomy of SFUDA is generally rough. And a large +number of relevant studies have emerged in the past year, +but the related papers are not included in that survey. In +addition, their work does not cover commonly used datasets +in this research field. +To fill the gap, in this paper, we provide a timely +and thorough literature review of existing deep learning +studies on source-free unsupervised domain adaptation. +Our goal is to cover SFUDA studies of the past few years +and provide a detailed and systematic SFUDA taxonomy. +Specifically, we classify existing SFUDA approaches into +two broad categories: (1) white-box SFUDA as shown in +Fig. 1 (b) and (2) black-box SFUDA as illustrated in Fig. 1 (c). +The difference between them lies in whether the model +parameters of the pre-trained source model are available +or not. Based on different learning strategies they use, we +further subdivide white-box and black-box SFUDA methods +arXiv:2301.00265v1 [cs.CV] 31 Dec 2022 + +SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY +2 +(a) Conventional UDA +(b) White-box SFUDA +Unlabeled +Target +Data +UDA +Tunable Source Parameters +Source Model +(c) Black-box SFUDA +API +UDA +Untunable Source Parameters +Source Model +UDA +Labeled +Source +Data +Fig. 1. Illustration of (a) conventional unsupervised domain adaptation (UDA), (b) white-box source-free UDA (SFUDA), and (c) black-box SFUDA. +Compared with (a) conventional UDA that relies on labeled source data {XS, YS} and unlabeled target data XT , (b, c) SFUDA performs knowledge +transfer by directly leveraging a pre-trained source model ΦS and unlabeled target data XT . The difference between (b) white-box SFUDA and (c) +black-box SFUDA lies in whether the learnable parameters of the source model ΦS are accessible or not. API: application programming interface. + Black-Box SFUDA +Self-Supervised Knowledge Distillation +Pseudo-Label Denoising + White-Box SFUDA +Model Fine-Tuning +Semi-Supervised Knowledge Distillation +Domain Alignment via Statistics +Contrastive Learning +Uncertainty-Guided Adaptation +Hidden Structure Mining +Source-Free +Unsupervised +Domain Adaptation +(SFUDA) +Data Generation +Domain Image Generation +Domain Distribution Generation +Future Outlook +Multi-Source/Target Domain Adaptation +Test-Time Domain Adaptation +Open/Partial/Universal-Set Domain Adaptation +Flexible Target Model Design +Cross-Modality Domain Adaptation +Continual/Lifelong Domain Adaptation +Semi-Supervised Domain Adaptation +Generative Distribution Alignment +Fig. 2. Taxonomy of existing source-free unsupervised domain adaptation (SFUDA) methods, as well as future outlook. +into finer categories, and the overall taxonomy is shown in +Fig. 2. Moreover, we discuss the challenges and insight for +methods in each category, provide a comprehensive compar- +ison between white-box and black-box SFUDA approaches, +summarize commonly used datasets in this field as well +as popular techniques to improve model generalizability +across different domains. We have to point out that SFUDA +is still under vigorous development, so we further discuss +the main challenges and provide insights into potential +future directions accordingly. +The rest of this survey is organized as follows. Section 2 +and Section 3 review existing white-box and black-box +SFUDA methods, respectively. In Section 4, we compare +white-box and black-box SFUDA and present useful strate- +gies to improve model generalization. Section 5 discusses +challenges of existing studies and future research directions. +Finally, we conclude this paper in Section 6. +2 +WHITE-BOX +SOURCE-FREE +UNSUPERVISED +DOMAIN ADAPTATION +Denote ΦS as the source model well-trained based on the +labeled source domain {XS, YS}, where XS and YS repre- +sent source data and the corresponding label information, +respectively. Denote {XT } as the unlabeled target domain +with only target samples XT . The goal of SFUDA is to learn +a target model ΦT for improved target inference based on +the pre-trained source model ΦS and unlabeled target data +XT . In the setting of white-box source-free domain adaptation, +the source data (i.e., XS and YS) cannot be accessed but the +training parameters of the source model ΦS are available. As +shown in the upper middle of Fig. 2, existing white-box +SFUDA studies can be divided into two categories: Data +Generation Method and Model Fine-Tuning Method, with +details elaborated as follows. +2.1 +Data Generation Method +2.1.1 +Domain Image Generation +Many studies aim to generate source-like image data and +achieve cross-domain adaptation by readily applying stan- +dard unsupervised domain adaptation techniques. Based +on different image generation strategies, these studies can +be divided into the following three subcategories: (1) batch +normalization statistics transfer, (2) surrogate source data +construction, and (3) generative adversarial network (GAN) +based image generation. +(1) Batch Normalization Statistics Transfer. Consider- +ing that batch normalization (BN) stores the running mean +and variance for a mini-batch of training data in each layer +of a deep learning model, some studies [26]–[28] explicitly +leverage such BN statistics for image style transfer, as il- +lustrated in Fig. 3. For instance, Yang et al. [26] generate +source-like images via a two-stage coarse-to-fine learning +strategy. In the coarse image generation step, BN statistics +stored in the source model are leveraged to preserve the +style characteristics of source images and also maintain the +content information of target data. In the fine image genera- +tion step, an image generator based on Fourier Transform +is developed to remove ambiguous textural components +of generated images and further improve image quality. +With generated source-like images and given target images, +a contrast distillation module and a compact consistency +measurement module are designed to perform feature-level +and output-level adaptation, respectively. Similarly, Hou et +al. [27] perform style transfer by matching BN statistics of +generated source-style image features with those saved in + +(Xs,Ys]X +TΦ +SSOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY +3 +Target Data +Noise +Source Model +Source Model +Source-like Data +UDA +Style Transfer via BN matching +& Content Preservation +Fig. 3. Illustration of Batch Normalization Statistics Transfer methods +for source image generation. By matching batch normalization (BN) +statistics between the upper and lower branches, source-like data can +be generated by preserving the target content but with source style. Un- +supervised domain adaptation (UDA) can then be performed between +source-like data and target data. +the pre-trained source model for image translation. Hong et +al. [28] generate source-like images by designing a style- +compensation transformation architecture guided by BN statis- +tics stored in the source model and the generated reliable +target pseudo-labels. +(2) Surrogate Source Data Construction. To compensate +for the inaccessible source domain, some studies [29]–[33] +construct surrogate/proxy source data by selecting appropriate +samples from the target domain directly, as illustrated in Fig. 4. +For example, Tian et al. [29] construct pseudo source sam- +ples directly from the provided target samples under the +guidance of a designed sample transport rule. The adaptation +step and sample transport learning step are performed alter- +nately to refine the approximated source domain and attain +confident labels for target data, thus achieving effective +cross-domain knowledge adaptation. Ding et al. [30] build a +category-balanced surrogate source domain using pseudo- +labeled target samples based on a prototype similarity mea- +surement. During model adaptation, intra-domain and inter- +domain mixup regularizations are introduced to transfer +label information from the proxy source domain to the target +domain, as well as simultaneously eliminate negative effects +caused by noisy labels. Ye et al. [31] select target samples +with high prediction confidence to construct a virtual source +set that mimics source distribution. To align the target and +virtual domains, they develop a weighted adversarial loss +based on distribution and an uncertainty measurement to +achieve cross-domain adaptation. Moreover, an uncertainty- +aware self-training mechanism is proposed to iteratively +produce the pseudo-labeled target set to further enhance +adaptation performance. Du et al. [32] construct a surrogate +source domain by first selecting target samples near the +source prototypes based on an entropy criterion, and then +enlarging them by a mixup augmentation strategy [34]. +The adversarial training is then used to explicitly mitigate +cross-domain distribution gap. Yao et al. [33] simulate proxy +source domain by freezing the source model and minimiz- +ing a supervised objective function for optimization. For the +simulated source set, global fitting is enforced by a model +gradient based equality constraint, which is optimized by an +alternating direction method of multipliers algorithm [35]. +(3) GAN-based Image Generation. Instead of approx- +Target Data +Surrogate Source +Data Construction +Surrogate Source Data +UDA +Fig. 4. Illustration of Surrogate Source Data Construction methods for +source data generation. These methods first construct surrogate/proxy +source data by selecting appropriate samples from the target domain +and then perform standard unsupervised domain adaptation (UDA). +imating the source domain directly using existing target +data, Kurmi et al. [36] simulate the source data by training +a GAN-based generator, as illustrated in Fig. 5. Specifically, +they first use a parametric conditional GAN to generate la- +beled proxy source data by treating the source classifier as +an energy based function. Then, they learn feature patterns +that are invariant across two domains via standard adver- +sarial learning for further adaptation. Hou et al. [37] also +update an image generator framework but they aim to translate +target images into the source-style ones instead of using the +latent noise as in [36]. In their method, the knowledge adap- +tation is achieved by training 1) a knowledge distillation loss +that mitigates the difference between features of newly gen- +erated source-style images and those of target images, and +2) a relation-preserving loss that maintains channel-level +relationship across different domains. Li et al. [38] propose a +GAN-embedded generator conditioned on a pre-defined label +to generate target-style data. By incorporating real target +samples, the learnable parameters of the generator and the +adapted model can be updated in a collaborative manner. +Moreover, two constraints, i.e., weight regularization and +clustering-based regularization, are utilized during model +adaptation to preserve source knowledge and ensure high- +confident target prediction, respectively. +2.1.2 +Domain Distribution Generation +Instead of generating source-like images directly, some stud- +ies propose to align feature prototypes or feature distribu- +tion of source data [39]–[43] with those in the target domain. +Specifically, Qiu et al. [39] generate feature prototypes for +each source category based on a conditional generator and +produce pseudo-labels for the target data. The cross-domain +prototype adaptation is achieved by aligning the features +derived from pseudo-labeled target samples to source pro- +totype with the same category label via contrastive learn- +ing. Tian et al. [40] construct a virtual domain by sim- +ply sampling from an approximated gaussian mixture model +(GMM) to mimic unseen source domain distribution. In +terms of adaptation procedure, they reduce the distribution +gap between the constructed virtual domain and the target +domain via adversarial training, thus bypassing inaccessible +source domain. Their practice is based on the assumption +that the feature prototype of each category can be mined +from each row of the source classifier’ weights [44]. With +the same assumption, Ding et al. [41] leverage such source +classifier weights and reliable target pseudo-labels derived +by spherical k-means clustering to estimate source feature +distribution. After that, proxy source data can be sampled +from the estimated source distribution, and a conventional +domain adaptation strategy [45] is used to explicitly perform + +X +TΦ +SX +TSOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY +4 +GAN Generator +0 +UDA +Generated +Source Data +Target Data +Noise +Pre-defined Label +Source Model +Fig. 5. +Illustration of Generative Adversarial Network (GAN) based +Image Generation methods for source data generation. Typically, a pre- +defined label and random noise act as the inputs of a GAN-based gen- +erator. By utilizing the pre-trained source model, they synthesize source +data for cross-domain adaptation. LCE: Cross-entropy loss function. +cross-domain feature distribution alignment. Stan et al. [42], +[43] propose to first generate a prototypical distribution +representing the source data in an embedding feature space +via GMM, and then perform source-free adaptation by +enforcing distribution alignment between source and target +domains via sliced Wasserstein distance [46]. +2.1.3 +Challenges and Insight +We classify existing domain image generation methods for +SFUDA into three subcategories. We present the challenges +of methods in each subcatetory and our insights below. +• Among the above-mentioned three subcategories, the first +one (i.e., batch normalization statistics transfer) explicitly +performs BN statistics matching between source and tar- +get domains for style transfer. Since the BN statistics +of the source model are off-the-shelf, these methods are +generally efficient and don’t require complex model train- +ing. However, BN statistics mainly focus on keeping the +style features while the content information cannot be +well preserved. Therefore, this strategy is more applicable +to scenarios where the contextual structure of images +between source and target domains does not differ too +much. It may not show good adaptation performance, +e.g., from a natural image to a cartoon image, since the +content information has significant changes. Note that BN +statistics transfer can also be used as a pre-processing step +in source-free domain adaptation, and it can be combined +with other strategies, e.g., circular learning [28], for more +effective knowledge transfer. +• Methods in the second subcategory (i.e., surrogate source +data construction) aim to approximate the proxy source +domain using appropriate target samples directly, fol- +lowed by conventional unsupervised domain adaptation. +Their application is quite broad, including semantic seg- +mentation [31], object recognition [30], [32], [33], image +classification [29], and digital recognition [29], [32]. In +general, methods in this group are straightforward and +computation-efficient by avoiding introducing extra hy- +perparameters, which is different from generative models. +However, because the proxy source samples are directly +selected from the target domain, these generated source +data may not effectively represent the original source +domain. Moreover, how to effectively select informative +target data for source data approximation is an important +topic to be investigated. Some studies have proposed var- +ious strategies for target data selection based on entropy +measurement [31], source prototype [30], [32], aggregated +source decision boundary [29], and equality constrained +optimization [33]. This is still an open but very interesting +future direction. For multi-source settings, it is promising +to study which source predictor(s) we should refer to for +effective target data selection. +• Methods in the third category (i.e., GAN-based image gen- +eration) typically synthesize images based on a generative +model. Since the generator can model underlying complex +distribution of source data with given random noise, +GAN-based models generally create more diverse images +compared with methods in second category (i.e., surro- +gate source data construction). However, these methods +introduce additional frameworks and learnable parame- +ters (e.g., generators and discriminators), which may cost +more computation resources. By comparing experimental +results, we find the surrogate source data construction +methods [32], [33] generally outperform the GAN-based +generators [36], [38]. The possible reason may be that the +constructed source data in the former are closer to real +data distributions, while those recovered in GAN-based +methods usually suffer from a mode collapse problem [30] +that leads to low-quality images. Note that the mode col- +lapse problem can be partly mitigated by using a carefully +tuned learning rate, manifold-guided training [47], and +virtual mapping [48], which is worth exploring further. +Different from image generation methods (Section 2.1.1) +that directly generate source/target-like images, the dis- +tribution generation methods (Section 2.1.2) generate fea- +ture prototype/distribution to achieve cross-domain feature +alignment. By comparing the reported experimental results, +we find that the distribution generation approaches [39]– +[41] usually outperform the GAN-based image generation +method [38]. And surrogate source data construction meth- +ods [30], [32] usually show superior performance compared +with the distribution generation methods [39], [40]. The +underlying reason could be that the source distributions +directly derived from the existing target data [30], [32] are +more accurate and stable than the approximated ones [39], +[40]. How to drive the approximated source distribution to +the real one can be further explored in the future. +2.2 +Model Fine-Tuning Method +Instead of generating source-like data for standard unsuper- +vised domain adaptation, many studies attempt to fine-tune +a pre-trained source model by exploiting unlabeled target +data in a self-supervised training scheme. Based on differ- +ent strategies for fine-tuning the source model, we divide +existing studies into five subcategories: (1) self-supervised +knowledge distillation, (2) domain alignment via statistics, +(3) contrastive learning, (4) uncertainty-guided adaptation, +and (5) hidden structure mining methods, as shown in +Fig. 2. More details are introduced in the following. +2.2.1 +Self-Supervised Knowledge Distillation +Many studies [22], [49]–[55] transfer knowledge learned +from source data to the target model via knowledge distil- +lation in a self-supervised manner, as illustrated in Fig. 6. In +these works, most of them [22], [49]–[52] achieve source-free +domain adaptation via a mean-teacher scheme for knowledge +transfer [56], where the target model not only learns from +unseen target domain but also well preserves source model + +X +TΦ +SLCESOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY +5 +Aug-α +Aug-β +Teacher Network +Student Network +EMA +LKD +Source Model +Initialize +Target Data +Initialize +Fig. 6. Illustration of Self-Supervised Knowledge Distillation methods +for source-free unsupervised domain adaptation. With target data from +different augmentations as inputs, a teacher-student framework is uti- +lized to exploit target features, where parameters of teacher network are +usually exponential moving average (EMA) of those of student network. +Aug-α and Aug-β denote two data augmentation methods (e.g., flip, +rotation, shift, noise addition, distortion, etc), respectively. LKD: Knowl- +edge distillation loss function. +information. For instance, Liu et al. [49] propose a self- +supervised distillation scheme for automatic polyp detec- +tion. By means of keeping output consistency of weak and +strong augmented polyp images, source knowledge is im- +plicitly transferred to the target model with a mean teacher +strategy [56]. Besides, a diversification flow paradigm is +designed to gradually eliminate the style sensitivity among +different domains, further enhancing model robustness to- +wards style diversification. Yang et al. [50] also propose +a self-supervised mean-teacher approach for knowledge +distillation, with a Transformer module [57] embedded. +This effectively helps the target model focus on object re- +gions rather than less informative background in an image, +thus improving model generalizability. Assuming that both +source and target images are generated from a domain- +invariant space by adding noise perturbations on each spe- +cific domain, Xiong et al. [51] establish a super target domain +via augmenting perturbations based on the original target +domain. The super and the original target domains are +fed into a mean-teacher framework, with three consistency +regularization terms (w.r.t. image, instance, and class-wise) +introduced for domain alignment. Chen et al. [22] first divide +the target data into clean and noisy subsets guided by a +computation loss and regard them as labeled and unlabeled +examples, and then utilize the mean teacher technique to +self-generate pseudo-labels for the unlabeled target data for +domain adaptation. +Instead of utilizing the conventional one-teacher one- +student paradigm, Liu et al. [52] construct a multi-teacher +multi-student framework, where each teacher/student net- +work is initialized using a public network pre-trained on +a single dataset. Here, a graph is constructed to model the +similarity among samples, and such relationship predicted +by the teacher networks is used to supervise the student net- +works via a mean-teacher technique. Rather than leverage +the mean-teacher paradigm that averages student’s weights, +Yu et al. [53] propose to distill knowledge from teacher to +student networks by style and structure regularizations, as +well as physical prior constraints. Instead of employing a +teacher-student network as the studies mentioned above, +Tang et al. [54] achieve data-free adaptation through gradual +knowledge distillation. Specifically, they first generate pseudo- +labels via a constructed neighborhood geometry, and then +Target Data +Source Model +Stored Source +Statistics +Target Model +Derived Target +Statistics +Statistics Discrepancy +Minimization +Fig. 7. Illustration of Domain Alignment via Statistics methods for +source-free unsupervised domain adaptation. The corresponding meth- +ods leverage batch statistics stored in the pre-trained source model +to approximate the distribution of inaccessible source data, and then +perform cross-domain adaptation by reducing distribution discrepancy +between source and target domains. +use pseudo-labels obtained from the latest epoch to super- +vise the current training epoch for knowledge transfer. +2.2.2 +Domain Alignment via Statistics +Many studies [58]–[64] leverage batch statistics stored in the +pre-trained source model to approximate the distribution +of inaccessible source data, and then perform cross-domain +adaptation by reducing distribution discrepancy between +source and target domains, as demonstrated in Fig. 7. For +example, Ishii et al. [58] approximate feature distribution +of inaccessible source data by using batch normalization +statistics (mean and variance) saved in the pre-trained source +model. Then, Kullback-Leibler (KL) divergence is utilized +to minimize the distributional discrepancy between source +and target domains, thus achieving domain-level alignment. +Inspired by [65], [66], Paul et al. [60] update the mean and +variance of BatchNorm [67] or InstanceNorm [68] of the +pre-trained model based on unseen target data. Not limited +to matching low-order batch-wise statistics (e.g., mean and +variance), Liu et al. [59] additionally incorporate high-order +batch-wise statistics, such as scale and shift parameters, to +explicitly keep cross-domain consistency. Moreover, they +quantify each channel’s transferability based on its inter- +domain divergence and assume that the channels with +lower divergence contribute more to domain adaptation. +Fan et al. [61] propose to align domain statistics adaptively by +modulating a learnable blending factor. By minimizing the +total objective function, each BN layer can dynamically ob- +tain its own optimal factor, which controls the contribution +of each domain to BN statistics estimation. The methods +mentioned above are all based on Gaussian-based statistics +domain alignment, while Eastwood et al. [62] attempt to +align histogram-based statistics of the marginal feature dis- +tributions of the target domain with those stored in the +pre-trained source model, thus well extending adaptation +to non-Gaussian distribution scenarios. +2.2.3 +Contrastive Learning +Many contrastive learning studies [19], [24], [69]–[72] per- +form data-free adaptation, which helps the target model +capture discriminative representations among unlabeled +target data. The main idea is to pull instances of similar +categories closer and push instances of different categories away +in feature space, as illustrated in Fig. 8. + +X +TΦ +SAug-QAug-βL +KDΦ +SX +TSOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY +6 +Before Adaptation +After Adaptation +Pull Close +Push Apart +Target Data +Fig. 8. Illustration of Contrastive Learning methods for source-free un- +supervised domain adaptation. These methods exploit discriminative +representations among unlabeled target data by pulling instances of +similar categories closer and pushing instances of different categories +away in feature space. +For instance, Xia et al. [69] first adaptively divide tar- +get instances into source-similar and source-dissimilar sets, +and then design a class-aware contrastive module for cross- +set distribution alignment. The idea is to enforce the com- +pactness of target instances from the same category and +reduce cross-domain discrepancy, thus prompting effective +knowledge transfer from the source model to target data. +Wang et al. [70] present a cross-domain contrastive learning +paradigm, which aims to minimize the distance between an +anchor instance from one domain and instances from other +domains that share the same category as the anchor. Due to +the unavailability of source data, they utilize source proto- +typical representations, i.e., weight vectors in the classifier +layer of a pre-trained source model, for feature alignment +across two domains. Huang et al. [19] tackle the data-free +domain adaptation by taking advantage of the historical +source hypothesis. Specifically, they propose a historical con- +trastive instance discrimination strategy to learn from target +samples by contrasting their learned embeddings generated +by the currently adapted and historical models. And they +also design a historical contrastive category discrimination +strategy to weight pseudo-labels of target data to learn +category-discriminative target representations, by calculat- +ing the consistency between the current and historical model +predictions. The two discrimination strategies help exploit +historical source knowledge, bypassing the dependence on +source data. Inspired by [73], Agarwal et al. [71] introduce +a pair-wise contrastive objective function to reduce intra- +category distance and meanwhile increase inter-category +distance based on generated target pseudo-labels. They also +introduce robust source and target models by taking advan- +tage of the generated adversarial instances, which facilitates +robust transfer of source knowledge to the target domain. +2.2.4 +Uncertainty-Guided Adaptation +Uncertainty can measure how well the target model fits the +data distribution [74], and many studies [75]–[82] utilize +such valuable information to guide target predictions in +source-free adaptation scenarios (see Fig. 9). +For instance, Fleuret et al. [75] estimate uncertainty based +on differences between predicted outputs with and without +Dropout operation [83]. By minimizing such differences, the +prediction uncertainty on target data is reduced, meanwhile +the learnable feature abstractor can be more robust to noise +perturbations. Lee et al. [76] exploit aleatoric uncertainty by +encouraging intra-domain consistency between target images +and their augmented ones and enforcing inter-domain feature +Target Data XT +Target Model +Uncertainty Measurement +Updated +e.g., Monte Carlo Dropout, +Entropy, Confidence, Consistency +Fig. 9. Illustration of Uncertainty-Guided Adaptation methods for source- +free unsupervised domain adaptation. These studies utilize uncertainty +to guide target predictions, and such valuable information can be mea- +sured by Monte Carlo Dropout, entropy, etc. +distribution consistency. Chen et al. [77] introduce a predic- +tion denoising approach for a cross-domain segmentation +task. In this study, a key component is introducing pixel- +wise denoising via uncertainty evaluation using Monte Carlo +Dropout [84], [85], which calculates the standard deviation +of several stochastic outputs and keeps it under a manually- +designed threshold. In this way, the noisy pseudo-labels +can be filtered out, helping improve pseudo-label quality +to achieve effective adaptation. Xu et al. [78] also propose an +uncertainty-guided pseudo-labeling denoising scheme, but +they use soft label correction instead of manually discarding +unreliable data points. Specifically, they first identify misla- +beled data points by utilizing a joint distribution matrix [86], +[87], and then assign larger confident weights to those with +higher certainty based on Monte Carlo Dropout. Combining +target data and the corresponding rectified pseudo-labeling, +a commonly used cross-entropy objective function can be +leveraged for training the target model. Sharing the similar +idea, Hegde et al. [79] allocate lower weights for uncer- +tain pseudo-labels, where the uncertainty is measured by +prediction variance based on Monte Carlo Dropout [84], +[85]. Considering that using Monte Carlo Dropout [84] +for uncertainty estimation requires manual hyperparameter +adjustment [88], Roy et al. [80] quantify source model’s un- +certainty using a Laplace approximation [89], [90]. For model +training, they assign smaller weights to those target samples +that are farther away from source hypothesis (measured by +uncertainty), avoiding misalignment of dissimilar samples. +Pei et al. [81] tackle the uncertainty issue from the perspec- +tive of improving source model transferability. Specifically, +they estimate channel-aware transferability of the source +model to target data based on an uncertainty distance, +which measures the closeness between target instances and +source distribution. With the aim of dynamically exploiting +the source model and target data, the target model obtains +the source knowledge from the transferable channels and +neglects those less-transferable ones. Unlike previous stud- +ies, Li et al. [82] quantify uncertainty using self-entropy and +propose a self-entropy descent mechanism to seek the optimal +confidence threshold for robust pseudo-labeling of target +data. They also leverage false negative mining and mosaic +augmentation [91] to further eliminate the negative influ- +ence of noisy labels to enhance adaptation performance. +2.2.5 +Hidden Structure Mining +Many studies [20], [92]–[98] take into consideration intrinsic +feature structures of target domain and update the target +model via clustering-aware pseudo-labeling. In Fig. 10, we +illustrate the main idea of hidden structure mining methods. + +SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY +7 +Before Adaptation +Class Centroid +Iteratively Update +Clustering Centroid +After Adaptation +Target Data +Fig. 10. Illustration of Hidden Structure Mining methods for source-free +unsupervised domain adaptation. These methods take into consider- +ation intrinsic feature structures of target domain and iterate between +target model refinement and clustering centroid update. +For example, Yang et al. [20] observe that target data can +intrinsically form a certain cluster structure that can be used +for domain adaptation. Specifically, they estimate affinity +among target data by taking into account the neighborhood +patterns captured from local-, reciprocal-, and expanded- +neighbors. Source-free adaptation is achieved by encour- +aging consistent predictions for those with high affinity. +Similarly, Yang et al. [92] also exploit neighborhood struc- +ture information of target data. They propose a local struc- +ture clustering strategy to encourage prediction consistency +among k-nearest target features, thus pushing target data +with semantically similar neighbors closer. Tang et al. [93] +leverage semantic constraints hidden in geometric structure +among target data to encourage robust clustering based +on a cognition mechanism [99]. Source hypothesis transfer +(SHOT) [94] and SHOT++ [95] attempt to mine the feature +structure of the target domain, but they cannot fully ex- +ploit the meaningful context since the used self-supervised +pseudo-labeling does not take into account each dimen- +sion’s covariance in the feature space. To address this issue, +Lee et al. [96] utilize GMM in the target domain to obtain +data structure, and design a joint model-data structure score to +concurrently exploit source and target knowledge. Yang et +al. [97] propose a novel neighborhood structure cluster- +ing method, which encourages intra-cluster target features +closer and meanwhile disperses those inter-cluster target +predictions far away. Li et al. [98] utilize neighbor structure +information from a new aspect by proposing a generic +and model smoothness-assisted Jacobian norm regularization +term, which is used to manipulate the consistency between +each target instance and its neighbors. This Jacobian norm +regularizer can be easily plugged into existing source-free +domain adaptation frameworks for boosting performance. +Different from the above mentioned methods, some +studies tackle source-free domain adaptation from other +perspectives. Li et al. [100] achieve data-free adaptation from +an adversarial-attack aspect, which aims to generate adversar- +ial target instances by adding diverse perturbations to attack +the target model. Then, mutual information maximization is +performed between representations extracted by the source +and target model for the same target instance. The above +two steps are performed alternatively, by which the domain- +invariant source knowledge can be preserved and the rich +target patterns can be well explored. Instead of explor- +ing domain-invariant features for cross-domain knowledge +transfer, Wang et al. [101] mine domain-invariant parameters +stored in the source model. They assume that only partial +domain-invariant parameters of the source model contribute +to domain adaptation, and their goal is to capture such pa- +rameters while penalizing the domain-specific ones. Liang et +al. [102] explore source-free adaptation from the perspec- +tive of minimum centroid shift, with the aim of searching a +subspace where target prototypes are mildly shifted from +source prototypes. An alternating optimization scheme is +leveraged for model convergence and target pseudo-label +update. Inspired by maximum classifier discrepancy [14], +Yang et al. [103] introduce an auxiliary bait classifier for cross- +domain feature alignment combined with the source anchor +classifier. These two classifiers aim to collaboratively push +uncertain target representations to the correct side of the +source classifier boundary. +2.2.6 +Challenges and Insight +We classify existing model fine-tuning methods for SFUDA +into five subcategories. The challenges of methods in each +subcatetory and our insights are presented below. +• The methods in the first subcategory, i.e., self-supervised +knowledge distillation, interpret source-free domain adap- +tation as a knowledge extraction and transfer process, +aiming to learn domain-invariant feature representations. +Most exiting studies transfer source knowledge to the +target model via a mean teacher strategy [56], where +teacher weights are an exponential moving average of +student weights. Hence, model parameters of both teacher +and student networks are tightly coupled, which may +lead to a performance bottleneck. A possible solution is to +introduce a dual-student framework and let one student +learn features flexibly, which may disentangle teacher- +student weights to some extent [104]. +• The second subcategory, i.e., domain alignment via statistics, +leverages batch statistics stored in a pre-trained source +model to approximate distribution of inaccessible source +data. Compared with other categories, these statistics- +based methods are lightweight and prone to generalize to +other tasks, since they require only a few update steps of +batch-wise statistics parameters and are potentially appli- +cable to real-time deployment [64]. However, they are not +suitable for problems that use deep network architectures +without batch normalization layers. +• The methods in the third subcategory, i.e., contrastive learn- +ing, aim to bring similar-class samples closer and push +dissimilar-class samples apart based on generated tar- +get pseudo-labels. Therefore, if the pseudo-labels contain +much noise, these methods may suffer from substantial +performance degradation. Moreover, a memory bank is +usually required to store the similarity relationship be- +tween current and historical feature representations of +target data, which could bring memory burden. It is +interesting to investigate the storage- and transmission- +efficient contrastive learning strategies in source-free set- +tings. In addition, several recent studies [105], [106] +have shown that data pair construction is crucial for +effective contrastive learning. One solution is utilizing +contrastive information between target data and their +augmented ones. Previous studies [107] often use either +strong or weak transformations for data augmentation, +where strong augmentations mostly distort the structures +of original images (e.g., shape distortion) while weak +augmentations usually limit transformations to preserve +the images’ structures (e.g., flip). Here we propose to + +SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY +8 +dynamically mix strong and weak augmentation of target +data, which may help learn more robust representations. +• The methods in fourth subcategory, i.e., uncertainty-guided +adaptation, focus on reducing prediction uncertainty of tar- +get data. Many studies [77], [78] use Monte Carlo Dropout +for uncertainty estimation, but this technique requires +specialized network architecture design and model train- +ing, bringing troublesome hyperparameter tuning [88]. A +recent study [77] points out that their method can only +handle problems with minor domain shift, and performs +poorly on problems with severe domain shift. It is inter- +esting to explore this challenging problem in the future. +• The last subcategory, i.e., hidden structure mining, considers +intrinsic clustering structure of the target domain, assum- +ing that geometric structure of target data may provide +informative context [93]. The advantage of these methods +is that no auxiliary frameworks are required, and thus, +they can be easily incorporated into other adaptation +frameworks. However, these methods have at least three +disadvantages. (1) Most existing studies need to iterate +between feature clustering and model update, which may +hinder training efficiency and cause a memory burden. +(2) These methods may be infeasible to handle extremely +large-scale datasets due to the difficulty of saving global +latent feature embeddings of the whole dataset [108]. +(3) Most studies construct target geometric structures in +Euclidean space, which may not be suitable for problems +with non-Euclidean data such as graphs. Thus, how to +improve training efficiency and deal with the large-size +dataset, as well as mining geometry information of non- +Euclidean data deserve further research. +From the application perspective, computation-efficient +approaches are more applicable for pixel-wise semantic +segmentation tasks, which require higher resources com- +pared with classification tasks. And those memory-intensive +approaches such as contrastive learning may be not suitable +for semantic segmentations. Moreover, it is worth noting +that data generation methods detailed in Section 2.1 can +be used in conjunction with the model fine-tuning methods +described in this section. For instance, one can first generate +a virtual source domain by selecting appropriate target +samples, and thus a standard unsupervised domain adap- +tation framework could be applied. To further exploit target +information, we then take account of geometric structure of +target samples and generate corresponding target pseudo- +labels to fine-tune the target model. These two steps can be +optimized iteratively, helping generate more representative +source domain and refine the target model. +3 +BLACK-BOX +SOURCE-FREE +UNSUPERVISED +DOMAIN ADAPTATION +Different from white-box methods, in the setting of black-box +source-free domain adaptation, both the source data {XS, YS} +and detailed parameters of the source model ΦS are not accessi- +ble. Only the hard or soft model predictions of the target +data XT from the source model ΦS are leveraged for do- +main adaptation. Depending on the utilization of the black- +box predictor, the existing black-box SFUDA studies can +be mainly divided into three categories: Self-Supervised +Knowledge Distillation, Pseudo-Label Denoising, and +Generative Distribution Alignment methods, with details +introduced below. +3.1 +Self-Supervised Knowledge Distillation +Some studies [109]–[114] construct a teacher-student-style +network architecture with knowledge distillation to trans- +late the source knowledge to the target domain in a self- +supervised manner. For instance, Liang et al. [109], [110] +enforce output consistency between a source model (i.e., +teacher) and a customized target model (i.e., student) via +a self-distillation loss. Specifically, a memory bank is first +constructed to store the prediction of each target sample +based on the black-box source model. This source model +then acts as a teacher to maintain an exponential moving +averaging of source and target prediction following [115], +[116]. Additionally, structural regularization on the target +domain is further incorporated during adaptation for more +effective knowledge distillation. Similarly, Liu et al. [111], +[112] employ an exponential mixup decay scheme to explicitly +keep prediction consistency of source and target domains, +thus gradually capturing target-specific feature representa- +tions and obtaining the target pseudo-labels. Xu et al. [113] +extend the teacher-student paradigm from image analysis +to more challenging video analysis, where not only spa- +tial features but also temporal information are taken into +consideration during domain adaptation. For knowledge +distillation, the target model is regarded as a student, which +aims to learn similar predictions generated by a teacher (i.e., +source) model. The teacher model is meanwhile updated +to maintain an exponential moving averaging prediction. +Instead of distilling knowledge between source and target +domains, Peng et al. [114] transfer knowledge between the +target network and its subnetwork in a mutual way, where +the subnetwork is a slimmer version generated from the +original target network following Yang et al. [117]. And +target features are extracted by leveraging multi-resolution +input images, helping improve the generalization ability of +the target network. Moreover, a novel data augmentation +strategy, called frequency MixUp, is proposed to empha- +size task-related regions-of-interests while simultaneously +reducing background interference. +3.2 +Pseudo-Label Denoising +Some studies [118], [119] tackle domain shift by carefully +denoising unreliable target pseudo-labels. For example, +Zhang et al. [118] combat noisy pseudo-labels via noise rate +estimation, which first preserves more training samples at +the start of the training process following [120] and then +gradually filters out the noisy ones based on their loss +values as training proceeds. The pseudo-labels are itera- +tively refined according to a category-dependent sampling +strategy, encouraging the model to capture more diverse +representations to improve model generalization ability. Dif- +ferent from Zhang et al. [118] that only select part of reliable +target data during model training, Luo et al. [119] take +into account all target data and rectify noisy pseudo-labels +from a negative learning aspect. Specifically, their approach +assigns complementary ground-truth labels for each target +sample, helping alleviate error accumulation for noisy pre- +diction. Moreover, a maximum squares objective function is + +SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY +9 +utilized as confidence regularization to prevent the target +model from being trapped in easy sample training. Yang et +al. [121] incorporate pseudo-label denoising and self-supervised +knowledge distillation into a unified framework. Specifically, +domain knowledge is first distilled from the trained source +predictor to warm up the target model by an exponential +moving averaging scheme. The unlabeled target domain +is then split into two subsets (i.e., easy and hard groups) +according to their adaptation difficulty [122], and the Mix- +Match strategy [123] is leveraged to progressively exploit all +target representations. In this way, the noise accumulation +is further suppressed, thereby improving the efficacy of +pseudo-label denoising. +3.3 +Generative Distribution Alignment +Different from the above methods, some studies perform +distribution alignment across domains in a generative way. +For instance, Yeh et al. [124] perform domain adaptation +by maximizing the lower bound in variational inference. +Specifically, they construct a generation path as well as an +inference path, where the generation path produces a prior +feature distribution derived from predicted category labels, +and the inference path approximates a posterior feature +distribution based on each target instance. The latent dis- +tribution alignment can be achieved by maximizing the evi- +dence lower vound in variational inference for cross-domain +adaptation. Similarly, Yang et al. [125] also construct the +generation and inference paths, but they achieve adaptation +via minimizing the upper bound of the prediction error of +target data in variational inference. Zhang et al. [126] achieve +source-free adaptation by first building multiple source +models and then generating a virtual intermediate surrogate +domain to select target samples with minimum inconsistency +predicted by the source models. The knowledge transfer +is achieved by feature distribution alignment between the +virtual surrogate domain and the target domain based on a +joint probability maximum mean discrepancy [127]. +3.4 +Challenges and Insight +In this section, we classify existing black-box SFUDA meth- +ods into three categories based on how they utilize the noisy +target predictions. The challenges of each category and our +insights are presented below. +• The first category, i.e., self-supervised knowledge distillation, +aims to gradually transfer source knowledge to a cus- +tomized target model by enforcing output consistency +between a teacher (source) and a student (target) network. +This learning strategy has also been used in white-box +SFUDA (see Section 2.2.1). The difference is that model +weights of student networks are accessible in white-box +SFUDA methods, but not in black-box SFUDA. In black- +box SFUDA, instead of leveraging any parameter details, +the teacher network is only updated by source predic- +tions and historical target predictions. The two items are +typically weighted by a momentum factor, which helps +dynamically adjust their contributions. Self-supervised +knowledge distillation has shown promising performance +in object recognition [109], semantic segmentation [111], +and video action recognition tasks [113]. +• The methods in the second category, i.e., pseudo-Label +denoising, tackle black-box SFUDA from the perspective of +noisy label rectification. It has shown that a pseudo-label +denoising approach [118] has inferior performance than +the self-supervised knowledge distillation method [109]. +The reason may be that the former [118] only focuses on +noisy prediction itself while neglecting target data struc- +ture that is well considered in the latter [109]. Considering +that pseudo-label denoising methods can tackle unbal- +anced label noise via noise rate estimation, combining +pseudo-label denoising with self-supervised knowledge +distillation strategies will be a promising future direc- +tion, especially in class-imbalance scenarios. Moreover, +if the black-box predictor only provides one-hot hard +predictions instead of probability predictions, the utility +of methods in this subcategory will be greatly reduced. +The reason is that the noise rate cannot be well estimated +in practice, e.g., there is nearly no difference between the +output of [0.45, 0.55] and that of [0.05, 0.95] because the +source predictor produces the same output (i.e., [0, 1]). +• The third category, i.e., generative distribution alignment, at- +tempts to perform domain adaptation by minimizing fea- +ture distribution discrepancy across the source and target +domains. Since source distribution is inaccessible in black- +box models, some generative approaches are utilized to +generate such reference distribution for target data to +align with, including variational autoencoder [124], [125] +and surrogate source domain construction [126]. These +methods are more suitbale for recognition/classification +tasks, but less suitable for semantic segmentation tasks. +For example, generating surrogate feature distribution of +an object (e.g., car) is usually easier than that of a semantic +scene (e.g., cityscape), since the latter contains different +objects and thus the pixel-wise neighborhood relationship +is difficult to model in practice. +Besides the strategies proposed above, it is also crucial +to build a general and robust black-box source model, with +which the target predictions tend to be more accurate. +To achieve that, one possible solution is augmenting the +diversity of source data (e.g., adding some perturbation) +before constructing the source model, which may eliminate +style discrepancy between two domains, thus improving the +generalization ability of the source model. Another solution +is using soft probability labels instead of hard one-hot +labels (e.g., [0, 1]) for model training, which prevents the +source model from being over-confident and helps enhance +its generalizability. Compared to white-box methods, there +are relatively few black-box SFUDA methods as well as +benchmark datasets, which needs to be further explored. +4 +DISCUSSION +In this section, we first compare the white-box and black- +box SFUDA methods and then summarize several useful +strategies to improve model generalizability. We also list +datasets commonly used in the field in Table 1. +4.1 +Comparison of White-Box and Black-Box SFUDA +By comparing existing white-box and black-box SFUDA +methods, we have the following interesting observations. + +SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY +10 +TABLE 1 +Commonly used datasets for evaluating the performance of source-free unsupervised domain adaptation (SFUDA) approaches. +Dataset +Domain # +Instance # +Category # +Description +Digit Recognition +Digits-Five [128] +5 +215,695 +10 +MNIST [129], SVHN [130], USPS [131], MNIST-M [13], Synthetic Digits [13] +Semantic Segmentation +Segmentation datasets +4 +45,766 +- +GTA5 [132], Cityscapes [133], SYNTHIA [134], NTHU [135] +Object Recognition +Office-31 [136] +3 +4,652 +31 +Amazon, Webcam, DSLR +Office-Home [137] +4 +15,500 +65 +Artistic, Clip Art, Product, Real-World +VisDA [138] +2 +280,000 +12 +Synthetic and real images +Office-Caltech-10 [139] +4 +2,533 +10 +Amazon, DSLR, Webcam, Caltech10 +ImageCLEF-DA [140] +4 +2,400 +12 +Caltech-256 [141], ImageNet ILSVRC2012 [16], PASCAL VOC2012 [142], Bing [143] +PACS [7] +4 +9,991 +7 +Art painting, Cartoon, Photo, Sketch +DomainNet [144] +6 +600,000 +345 +Clipart, Infograph, Painting, Quickdraw, Real, Sketch +MiniDomainNet [145] +4 +140,000 +126 +Clipart, Painting, Real, Sketch +PointDA-10 [146] +3 +33,067 +10 +ModelNet [147], ShapeNet [148], ScanNet [149] +Face Anti-Spoofing +Face datasets +4 +7,130 +- +Replay-Attack [150], OULU-NPU [151], CASIA-MFSD [152], MSU-MFSD [153] +LiDAR Detection +LiDAR datasets +3 +158,510 +- +Waymo [154], KITTI [155], nuScenes [156] +Video Action Recognition +UCF-HMDBfull [157] +2 +3,209 +12 +UCF101 [158], HMDB51 [159] +Sports-DA [160] +3 +40,718 +23 +UCF10 [158], Sports-1M [161], Kinetics [162] +Daily-DA [160] +4 +18,949 +8 +ARID [163], HMDB51 [159], Moments-in-Time [164], Kinetics [162] +Traffic Sign Recognition +Sign datasets +2 +151,839 +43 +Syn.Signs [165], GTSRB [166] +Image Classification +VLCS [167] +4 +10,729 +5 +Caltech101 [168], LabelMe [169], SUN09 [170], VOC2007 [142] +Medical Data +BraTS2018 [171] +2 +285 +- +Cross-disease (high and low grade glioma), cross-modality (T1, T1ce, T2, FLAIR) +MMWHS [172] +2 +40 +- +Cross-modality (magnetic resonance imaging, computed tomography) +Brain skull stripping [173] +3 +35 +- +NFBS [174], ADNI [175], dHCP [176] +Polyp segmentation +4 +2,718 +- +CVC-ClincDB [177], Abnormal Symptoms [178], ETIS-Larib [179], EndoScene [180] +EEG MI Classification [126] +4 +528 +2/4 +MI2-2 [181], MI2-4 [181], MI2015 [182], AlexMI [183] +Prostate segmentation +2 +682 +- +NCI-ISBI 2013 Challenge [184], PROMISE12 challenge [185] +Optic disc&cup segmentation +3 +660 +- +REFUGE [186], RIMONE-r3 [187], Drishti-GS [188] +Autism diagnosis +4 +411 +2 +NYU, USM, UCLA, and UM of ABIDE dataset [189] +• Compared with black-box SFUDA that cannot access any +source parameters, white-box SFUDA is capable of mining +more source knowledge (e.g., batch statistics) that facili- +tates more effective domain adaptation. +• White-box SFUDA methods may suffer from data pri- +vacy leakage problems [118]. For instance, Yin et al. [190] +reveal that raw data can be recovered based on source +image distribution via a deep inversion technique. Using +a membership inference attack strategy [191], [192], it is +possible to infer whether a given sample exists or not +in training dataset, thereby revealing private information. +The black-box SFUDA can help protect data privacy be- +cause only application programming interface (API) is +accessible while detailed model weights are withheld, +but it may suffer from performance degradation of cross- +domain adaptation. +• Most white-box SFUDA methods assume that model ar- +chitecture is shared between source and target domains, +while the black-box SFUDA methods try to design task- +specific target models for knowledge transfer. Such flex- +ible model design in black-box SFUDA methods is very +useful for target users with low computation resources, +since they can design more efficient and lightweight target +models for domain adaptation. +• Black-box SFUDA methods neither require data synthe- +sis nor model fine-tuning, which helps to accelerate the +convergence process of model training. In contrast, white- +box methods are usually computationally intensive and +time-consuming. For instance, it is reported that the com- +putational cost of a black-box SFUDA method [126] is +0.83s while that of two competing white-box methods +are 3.17s [94] and 22.43s [193], respectively, reflecting the +computation efficiency of black-box SFUDA. +In summary, when using white-box and black-box +SFUDA methods, we have to make a trade-off between +obtaining better performance, protecting confidential infor- +mation, and reducing computational and memory costs. +4.2 +Useful Strategies for Improved Generalizability +To facilitate research practice in this field, we summarize +several useful techniques that could be used to improve the +generalizability of learning models for source-free unsuper- +vised domain adaptation. +4.2.1 +Entropy Minimization Loss +Most SFUDA methods utilize an entropy minimization +loss [194] to reduce uncertainty of model predictions [27], +[59], [75], [94], [111], [112], [195]–[200]. This simple yet +effective strategy encourages the model to generate one-hot +predictions for more confident learning. + +SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY +11 +4.2.2 +Diversity Enforcing Loss +To prevent predicted labels from collapsing to categories +with larger number of samples, many studies leverage a +diversity enforcing loss to encourage diverse predictions +over target domain [80], [94], [196], [201]–[205] . The usual +practice is to maximize the entropy of empirical label distri- +bution over the batch-wise average of model predictions. +4.2.3 +Label Smoothing Technique +In source-free adaptation studies, a pre-trained source +model is generally obtained via training on labeled source +data before adaptation stages. Currently, many studies use +a label smoothing technique [206], [207] to produce a robust +source model [20], [30], [94], [101], [208], [209]. This tech- +nique aims to transform original training labels from hard +labels (e.g., 1) to soft labels (e.g., 0.95), which prevents the +source model from being over-confident, helping enhance +its generalization ability. Also, the experiments have shown +that label smoothing can encourage closer representations +of training samples from the same category [206]. With a +more general and robust source model, it is likely to boost +adaptation performance on target domain. +4.2.4 +Model Regularization +Many regularization terms are utilized in existing SFUDA +methods by incorporating some prior knowledge. For in- +stance, an early learning regularization [39], [120], [210] is +used to prevent the model from over-fitting to label noise. A +stability regularization [38], [211]–[213] is leveraged to pre- +vent parameters of the target model to deviate from those +of the source model. A local smoothness regularization [38], +[214] is used to encourage output consistency between the +target model and its noise-perturbed counterpart, helping +improve robustness of the target model. A mixup regular- +ization [30], [109], [114], [215], [216] is used to enforce pre- +diction consistency between original and augmented data, +which can mitigate the negative influence of noisy labels. +4.2.5 +Confidence Thresholding +Many studies leverage pseudo-labeling to train the tar- +get model in a self-supervised way. Instead of utilizing a +manually-designed threshold to identify reliable/confident +pseudo-labels, a commonly used strategy is automatically +learning the confidence threshold for reliable pseudo-label +selection [217]. To further tackle the class-imbalance prob- +lem, some studies [75], [78], [212], [218], [219] propose to +learn dynamic threshold for each category, which provides +a fair chance for categories with limited samples to generate +pseudo-labels for self-training. +5 +FUTURE OUTLOOK +5.1 +Multi-Source/Target Domain Adaptation +To utilize diverse and rich information of multiple domains, +a few studies [29], [193], [204], [220], [221] propose multi- +source data-free adaptation to transfer source knowledge +to the target domain. Tian et al. [29] introduce a sample +transport learning method, but the proposed model is shal- +low, and thus cannot handle highly nonlinear feature ex- +traction. To tackle this problem, several deep learning based +models [193], [204] are proposed. But they ignore the fact +that the generated target pseudo-labels may be noisy, which +may cause training bias when matching target domains with +large domain gaps. The key to solving problems with multi- +source domains is quantifying the transferability of different +source models and utilizing their complementary informa- +tion for promoting cross-domain adaptation. Even several +strategies are proposed (e.g., aggregation weight [193] and +source-specific transferable perception [204]), more explo- +rations are encouraged to address the problem of negative +transfer during cross-domain knowledge transfer. +A few studies [222]–[226] incorporate federated learning +into domain adaptation scenarios. Federated learning [227]– +[229] is a decentralized scheme to facilitate collaborative +learning among multiple distributed clients without shar- +ing training data or model parameters. The constraint that +prevents data and parameter transmission across different +source domains is not required in multi-source-free domain +adaptation. For instance, federated adversarial domain adapta- +tion (FADA) introduced by Peng et al. [222] is among the +first attempts to propose the concept of federated domain +adaptation, which employs a dynamic attention mecha- +nism to transfer knowledge from multi-source domains to +an unlabeled target domain. In this method, each source +model needs to synchronize with the target domain after +each training batch, resulting in huge computation costs +and potential risk of privacy leakage [230]. To tackle this +problem, Feng et al. [223] introduces a consensus focus +schema that greatly improves communication efficiency for +decentralized domain adaptation. Moreover, Song et al. [225] +utilize a homomorphic encryption approach for privacy pro- +tection, and Qin et al. [226] introduce a flexible uncertainty- +aware strategy for reliable source selection. However, cur- +rent federated learning studies usually produce a common +model for all clients without considering heterogeneity of +data distribution of different clients. Therefore, the common +model cannot adapt to each client adaptively, which may +affect adaptation performance. It would be very interesting +to investigate personalized federated learning [231], with +which current or new clients can easily adapt to their +own local dataset by performing a few optimization steps. +Besides, all the methods mentioned above require labeled +data from multiple sources to train a federated model, in- +evitably increasing annotation costs. Therefore, approaches +that effectively exploit unlabeled data from multiple source +domains in a decentralized way are urgently needed. +On the other hand, Yao et al. [232] and Shenaj et al. [233] +have proposed several federated multi-target domain adap- +tation strategies for transferring knowledge of a labeled +source server to multiple unlabeled target clients. More ad- +vanced techniques for federated multi-target domain adap- +tation are highly desirable, by considering computation +and communication cost, annotation burden, and privacy +protection of different target domains. +5.2 +Test-Time Domain Adaptation +Most SFUDA approaches require pre-collected unlabeled +target data for model training, termed “training-time adap- +tation”. Test-time adaptation [198], [234]–[237] has been +investigated by adapting the source model to the target + +SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY +12 +domain during inference procedure only. The advantages of +test-time adaptation are mainly twofold: (1) The adaptation +process does not need iterative training, which greatly im- +proves computational efficiency, so the model can be easily +deployed in an online manner. (2) Without relying on target +training data, test-time adaptation is expected to be well +generalized to diverse target domains. Even current studies +have made promising achievements, there are still some +problems worth exploring, listed as follows. +Some studies [198], [238], [239] need to access batch- +sized (>1) target samples during inference, which can- +not handle scenarios where target samples arrive one-by- +one sequentially. Two studies [240], [241] perform image- +wise adaptation rather than batch-wise adaptation, but they +cannot deal with cases with large distribution shift. It is +interesting to explore how to handle the scenarios where +test instances come from continuous changeable domains +in the future. Additionally, the solutions on how to adap- +tively exploit test data can be further explored [242], such +as adjusting model weights dynamically based on sample +discrepancy across domains. +5.3 +Open/Partial/Universal-Set Domain Adaptation +This survey focuses on close-set source-free domain adapta- +tion, where the label space of source and target domains is +consistent. But practical scenarios are much more compli- +cated when the category shift issue occurs across different +domains. There are three non-close-set scenarios: (1) open-set +(Cs ⊂ Ct) problems, (2) partial-set (Cs ⊃ Ct) problems, and +(3) universal-set (Cs\Ct ̸= ∅ and Ct\Cs ̸= ∅, Cs ⊂ Ct, +Cs ⊃ Ct) problems, where Cs and Ct denote the category +label set for source and target domains, respectively. +Currently, only a few studies [18], [243], [244] attempt +to handle the category shift problem in source-free adapta- +tion scenarios, including out-of-distribution data construc- +tion [18], [243], neighborhood clustering learning [245], +uncertainty-based progressive learning [246], and mutual +information maximization [203]. The main idea behind these +studies is to recognize out-of-source distribution samples +and improve generalization ability of the source model. +However, the model performance of existing studies is +not quite satisfactory due to the inaccessibility of valuable +category-gap knowledge. One possible solution is to adap- +tively learn a threshold instead of using a fixed one to +determine the acceptance/rejection of each target sample +as a “known” category via some similarity measurement. +Moreover, some strategies used in non-source-free domain +adaptation can also be borrowed, such as distribution +weighted combining rule [247], category-invariant represen- +tation learning [248], one-vs-all learning scheme [249], and +global-local optimization [250]. +5.4 +Flexible Target Model Design +For black-box SFUDA methods, due to unavailability of +structure and parameters of target model, one usually has +to manually design a target model. For instance, Liu et +al. [111] choose a U-Net based framework as the target +model for segmentation. However, such manually designed +architectures may be not suitable when adapting to the tar- +get domain. It is expected that the automatic design of target +models, e.g., using neural architecture search (NAS) [251]– +[253], helps improve the learning performance. Consider- +ing that NAS has recently become a popular strategy for +searching proper network architectures in deep learning, we +can integrate it into SFUDA scenarios to find more proper +and efficient target models. And how to balance the search +space and search cost of network parameters can be further +investigated. Moreover, hyperparameters used in NAS (e.g., +optimizer strategy, weight decay regularization) should be +carefully considered since they also have a significant im- +pact on network performance [254]. +5.5 +Cross-Modality Domain Adaptation +Existing studies mainly focus on one single modality for +domain adaptation, while a few studies perform cross- +modality adaptation in source-free settings [28], [255]. For +instance, for medical data analysis, the acquisition expense +of computed tomography (CT) scans is generally less than +that of magnetic resonance imaging (MRI) scans, hence it +may greatly reduce annotation cost for a segmentation task +when transferring a source model trained on CT images to +MRI scans [28]. Moreover, in computer vision field, it would +be promising to investigate cross-modality adaptation in +the future, e.g., image→video, which aims to achieve video +recognition based on the source model trained on the image +dataset. Also, how to effectively integrate multi-modality +(e.g., image, sound, text, and video) data for domain adapta- +tion in a source-free way is an interesting but not yet widely +studied problem. +5.6 +Continual/Lifelong Domain Adaptation +Most current studies focus on improving adaptation per- +formance on the target domain while neglecting the perfor- +mance on source domain, running the risk of catastrophic +forgetting problems [256]. To address this issue, several +solutions have been developed from different aspects, such +as domain expansion [257], historical contrastive learn- +ing [19], domain attention regularization [92], and model +perturbation [258], while there is still massive room for +performance improvement. Inspired by continual/lifelong +learning [259]–[262], continual domain adaptation has re- +cently made great progress by investigating gradient reg- +ularization [263], iterative neuron restoration [264], buffer +sample mixture [265], etc. The continual domain adaptation +in source-free settings for mitigation of catastrophic forget- +ting remains an underdeveloped topic that can be further +explored in the future. +5.7 +Semi-Supervised Domain Adaptation +Source-free domain adaptation in semi-supervised settings +(i.e., with a few labeled target data involved for model train- +ing) has also been explored in recent years [197], [266], [267]. +It usually performs semi-supervised adaptation with the +help of active learning [268], [269], model memorization rev- +elation [270], and consistency and diversity learning [271]. +There is still a lot of space for improvement with a limited +number of labeled target samples, e.g., by fine-tuning the +current source-free adaptation frameworks, but this is not +the focus of this survey. + +SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY +13 +6 +CONCLUSION +In this paper, we provide a comprehensive review of recent +progress in source-free unsupervised domain adaptation +(SFUDA). We classify existing SFUDA studies into white- +box and black-box groups, and each group is further catego- +rized based on different learning strategies. The challenges +of methods in each category and our insights are provided. +We then compare white-box and black-box SFUDA meth- +ods, discuss effective techniques for improving adaptation +performance, and summarize commonly used datasets. We +finally discuss promising future research directions. It is +worth noting that the research topic of source-free unsu- +pervised domain adaptation is still in its early stages, and +we hope this survey can spark new ideas and attract more +researchers to advance this high-impact research field. +ACKNOWLEDGMENT +This work was supported by NIH grants RF1AG073297 and +R01MH108560. +REFERENCES +[1] +A. Voulodimos, N. Doulamis, A. Doulamis, and E. Protopa- +padakis, “Deep learning for computer vision: A brief review,” +Computational Intelligence and Neuroscience, vol. 2018, 2018. +[2] +M. 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Wang, “Learning invariant rep- +resentation with consistency and diversity for semi-supervised +source hypothesis transfer,” arXiv preprint arXiv:2107.03008, 2021. + diff --git a/09AyT4oBgHgl3EQfbffL/content/tmp_files/load_file.txt b/09AyT4oBgHgl3EQfbffL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c476c3c444a288fe4edd417d8451939095ed23ba --- /dev/null +++ b/09AyT4oBgHgl3EQfbffL/content/tmp_files/load_file.txt @@ -0,0 +1,2974 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf,len=2973 +page_content='SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY 1 Source-Free Unsupervised Domain Adaptation: A Survey Yuqi Fang, Pew-Thian Yap, Senior Member, IEEE, Weili Lin, Hongtu Zhu, and Mingxia Liu, Senior Member, IEEE Abstract—Unsupervised domain adaptation (UDA) via deep learning has attracted appealing attention for tackling domain-shift problems caused by distribution discrepancy across different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Existing UDA approaches highly depend on the accessibility of source domain data, which is usually limited in practical scenarios due to privacy protection, data storage and transmission cost, and computation burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' To tackle this issue, many source-free unsupervised domain adaptation (SFUDA) methods have been proposed recently, which perform knowledge transfer from a pre-trained source model to unlabeled target domain with source data inaccessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' A comprehensive review of these works on SFUDA is of great significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In this paper, we provide a timely and systematic literature review of existing SFUDA approaches from a technical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Specifically, we categorize current SFUDA studies into two groups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', white-box SFUDA and black-box SFUDA, and further divide them into finer subcategories based on different learning strategies they use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' We also investigate the challenges of methods in each subcategory, discuss the advantages/disadvantages of white-box and black-box SFUDA methods, conclude the commonly used benchmark datasets, and summarize the popular techniques for improved generalizability of models learned without using source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' We finally discuss several promising future directions in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Index Terms—Domain adaptation, source-free, unsupervised learning, survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 1 INTRODUCTION D EEP learning, based on deep neural networks with rep- resentation learning, has emerged as a promising tech- nique and made remarkable progress over the past decade, covering the field of computer vision [1], [2], medical data analysis [3], [4], natural language processing [5], [6], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For problems with multiple domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', different datasets or imaging sites), the typical learning process of a deep neural network is to transfer the model learned on a source domain to a target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' However, performance degradation is often observed when there exists a distribution gap between the source and target domains, which is termed “domain shift” problem [7]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' To tackle this problem, various do- main adaptation algorithms [10], [11] have been proposed to perform knowledge transfer by reducing inter-domain distribution discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' To avoid intensive burden of data annotation, unsupervised domain adaptation has achieved much progress [12]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 1 (a), unsuper- vised domain adaptation aims to transfer knowledge from a labeled source domain to a target domain without accessing any target label information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Existing deep learning studies on unsupervised domain adaptation highly depend on the accessibility of source data, which is usually limited in practical scenarios due to the following possible reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' (1) Data privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Many source datasets containing confidential information, such as medical and facial data, are not available to third parties due to privacy and security protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' (2) Data storage and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Fang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Yap, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Lin and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Liu are with the Department of Radiology and Biomedical Research Imaging Center, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Zhu is with the Department of Biostatistics, University of North Carolina at Chapel Hill, NC 27599, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Corresponding author: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Liu (mxliu@med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' transmission cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The storage and transmission of large- scale source datasets, such as ImageNet [16], could bring much economic burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' (3) Computation burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Training on extremely large source datasets requires high computational resources, which is not practical, especially in real-time deployment cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Thus, there is a high demand for source- free unsupervised domain adaptation (SFUDA) methods that transfer a pre-trained source model to the unlabeled target domain without accessing any source data [17]–[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Many promising SFUDA algorithms have been devel- oped recently to address problems in the fields of seman- tic segmentation [21], image classification [22], object de- tection [23], face anti-spoofing [24], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' A comprehensive review of current studies on SFUDA as well as an outlook on future research directions are urgently needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [25] present a review on data-free knowledge transfer, where SFUDA only accounts for part of the review and the taxonomy of SFUDA is generally rough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' And a large number of relevant studies have emerged in the past year, but the related papers are not included in that survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In addition, their work does not cover commonly used datasets in this research field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' To fill the gap, in this paper, we provide a timely and thorough literature review of existing deep learning studies on source-free unsupervised domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Our goal is to cover SFUDA studies of the past few years and provide a detailed and systematic SFUDA taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Specifically, we classify existing SFUDA approaches into two broad categories: (1) white-box SFUDA as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 1 (b) and (2) black-box SFUDA as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The difference between them lies in whether the model parameters of the pre-trained source model are available or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Based on different learning strategies they use, we further subdivide white-box and black-box SFUDA methods arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='00265v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='CV] 31 Dec 2022 SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY 2 (a) Conventional UDA (b) White-box SFUDA Unlabeled Target Data UDA Tunable Source Parameters Source Model (c) Black-box SFUDA API UDA Untunable Source Parameters Source Model UDA Labeled Source Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Illustration of (a) conventional unsupervised domain adaptation (UDA), (b) white-box source-free UDA (SFUDA), and (c) black-box SFUDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Compared with (a) conventional UDA that relies on labeled source data {XS, YS} and unlabeled target data XT , (b, c) SFUDA performs knowledge transfer by directly leveraging a pre-trained source model ΦS and unlabeled target data XT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The difference between (b) white-box SFUDA and (c) black-box SFUDA lies in whether the learnable parameters of the source model ΦS are accessible or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' API: application programming interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Black-Box SFUDA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Self-Supervised Knowledge Distillation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Pseudo-Label Denoising ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='White-Box SFUDA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Model Fine-Tuning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Semi-Supervised Knowledge Distillation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Domain Alignment via Statistics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Contrastive Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Uncertainty-Guided Adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Hidden Structure Mining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Source-Free ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Unsupervised ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Domain Adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='(SFUDA) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Data Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Domain Image Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Domain Distribution Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Future Outlook ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Multi-Source/Target Domain Adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Test-Time Domain Adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Open/Partial/Universal-Set Domain Adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Flexible Target Model Design ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Cross-Modality Domain Adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Continual/Lifelong Domain Adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Semi-Supervised Domain Adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Generative Distribution Alignment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Taxonomy of existing source-free unsupervised domain adaptation (SFUDA) methods, as well as future outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' into finer categories, and the overall taxonomy is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moreover, we discuss the challenges and insight for methods in each category, provide a comprehensive compar- ison between white-box and black-box SFUDA approaches, summarize commonly used datasets in this field as well as popular techniques to improve model generalizability across different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' We have to point out that SFUDA is still under vigorous development, so we further discuss the main challenges and provide insights into potential future directions accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The rest of this survey is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Section 2 and Section 3 review existing white-box and black-box SFUDA methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In Section 4, we compare white-box and black-box SFUDA and present useful strate- gies to improve model generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Section 5 discusses challenges of existing studies and future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Finally, we conclude this paper in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2 WHITE-BOX SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION Denote ΦS as the source model well-trained based on the labeled source domain {XS, YS}, where XS and YS repre- sent source data and the corresponding label information, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Denote {XT } as the unlabeled target domain with only target samples XT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The goal of SFUDA is to learn a target model ΦT for improved target inference based on the pre-trained source model ΦS and unlabeled target data XT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In the setting of white-box source-free domain adaptation, the source data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', XS and YS) cannot be accessed but the training parameters of the source model ΦS are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' As shown in the upper middle of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2, existing white-box SFUDA studies can be divided into two categories: Data Generation Method and Model Fine-Tuning Method, with details elaborated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1 Data Generation Method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1 Domain Image Generation Many studies aim to generate source-like image data and achieve cross-domain adaptation by readily applying stan- dard unsupervised domain adaptation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Based on different image generation strategies, these studies can be divided into the following three subcategories: (1) batch normalization statistics transfer, (2) surrogate source data construction, and (3) generative adversarial network (GAN) based image generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' (1) Batch Normalization Statistics Transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Consider- ing that batch normalization (BN) stores the running mean and variance for a mini-batch of training data in each layer of a deep learning model, some studies [26]–[28] explicitly leverage such BN statistics for image style transfer, as il- lustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For instance, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [26] generate source-like images via a two-stage coarse-to-fine learning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In the coarse image generation step, BN statistics stored in the source model are leveraged to preserve the style characteristics of source images and also maintain the content information of target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In the fine image genera- tion step, an image generator based on Fourier Transform is developed to remove ambiguous textural components of generated images and further improve image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' With generated source-like images and given target images, a contrast distillation module and a compact consistency measurement module are designed to perform feature-level and output-level adaptation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Similarly, Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [27] perform style transfer by matching BN statistics of generated source-style image features with those saved in (Xs,Ys]X TΦ SSOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY 3 Target Data Noise Source Model Source Model Source-like Data UDA Style Transfer via BN matching & Content Preservation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Illustration of Batch Normalization Statistics Transfer methods for source image generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' By matching batch normalization (BN) statistics between the upper and lower branches, source-like data can be generated by preserving the target content but with source style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Un- supervised domain adaptation (UDA) can then be performed between source-like data and target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' the pre-trained source model for image translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [28] generate source-like images by designing a style- compensation transformation architecture guided by BN statis- tics stored in the source model and the generated reliable target pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' (2) Surrogate Source Data Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' To compensate for the inaccessible source domain, some studies [29]–[33] construct surrogate/proxy source data by selecting appropriate samples from the target domain directly, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For example, Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [29] construct pseudo source sam- ples directly from the provided target samples under the guidance of a designed sample transport rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The adaptation step and sample transport learning step are performed alter- nately to refine the approximated source domain and attain confident labels for target data, thus achieving effective cross-domain knowledge adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [30] build a category-balanced surrogate source domain using pseudo- labeled target samples based on a prototype similarity mea- surement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' During model adaptation, intra-domain and inter- domain mixup regularizations are introduced to transfer label information from the proxy source domain to the target domain, as well as simultaneously eliminate negative effects caused by noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [31] select target samples with high prediction confidence to construct a virtual source set that mimics source distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' To align the target and virtual domains, they develop a weighted adversarial loss based on distribution and an uncertainty measurement to achieve cross-domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moreover, an uncertainty- aware self-training mechanism is proposed to iteratively produce the pseudo-labeled target set to further enhance adaptation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [32] construct a surrogate source domain by first selecting target samples near the source prototypes based on an entropy criterion, and then enlarging them by a mixup augmentation strategy [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The adversarial training is then used to explicitly mitigate cross-domain distribution gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [33] simulate proxy source domain by freezing the source model and minimiz- ing a supervised objective function for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For the simulated source set, global fitting is enforced by a model gradient based equality constraint, which is optimized by an alternating direction method of multipliers algorithm [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' (3) GAN-based Image Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Instead of approx- Target Data Surrogate Source Data Construction Surrogate Source Data UDA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Illustration of Surrogate Source Data Construction methods for source data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' These methods first construct surrogate/proxy source data by selecting appropriate samples from the target domain and then perform standard unsupervised domain adaptation (UDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' imating the source domain directly using existing target data, Kurmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [36] simulate the source data by training a GAN-based generator, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Specifically, they first use a parametric conditional GAN to generate la- beled proxy source data by treating the source classifier as an energy based function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Then, they learn feature patterns that are invariant across two domains via standard adver- sarial learning for further adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [37] also update an image generator framework but they aim to translate target images into the source-style ones instead of using the latent noise as in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In their method, the knowledge adap- tation is achieved by training 1) a knowledge distillation loss that mitigates the difference between features of newly gen- erated source-style images and those of target images, and 2) a relation-preserving loss that maintains channel-level relationship across different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [38] propose a GAN-embedded generator conditioned on a pre-defined label to generate target-style data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' By incorporating real target samples, the learnable parameters of the generator and the adapted model can be updated in a collaborative manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moreover, two constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', weight regularization and clustering-based regularization, are utilized during model adaptation to preserve source knowledge and ensure high- confident target prediction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2 Domain Distribution Generation Instead of generating source-like images directly, some stud- ies propose to align feature prototypes or feature distribu- tion of source data [39]–[43] with those in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Specifically, Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [39] generate feature prototypes for each source category based on a conditional generator and produce pseudo-labels for the target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The cross-domain prototype adaptation is achieved by aligning the features derived from pseudo-labeled target samples to source pro- totype with the same category label via contrastive learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [40] construct a virtual domain by sim- ply sampling from an approximated gaussian mixture model (GMM) to mimic unseen source domain distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In terms of adaptation procedure, they reduce the distribution gap between the constructed virtual domain and the target domain via adversarial training, thus bypassing inaccessible source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Their practice is based on the assumption that the feature prototype of each category can be mined from each row of the source classifier’ weights [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' With the same assumption, Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [41] leverage such source classifier weights and reliable target pseudo-labels derived by spherical k-means clustering to estimate source feature distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' After that, proxy source data can be sampled from the estimated source distribution, and a conventional domain adaptation strategy [45] is used to explicitly perform X TΦ SX TSOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY 4 GAN Generator 0 UDA Generated Source Data Target Data Noise Pre-defined Label Source Model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Illustration of Generative Adversarial Network (GAN) based Image Generation methods for source data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Typically, a pre- defined label and random noise act as the inputs of a GAN-based gen- erator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' By utilizing the pre-trained source model, they synthesize source data for cross-domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' LCE: Cross-entropy loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' cross-domain feature distribution alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Stan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [42], [43] propose to first generate a prototypical distribution representing the source data in an embedding feature space via GMM, and then perform source-free adaptation by enforcing distribution alignment between source and target domains via sliced Wasserstein distance [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='3 Challenges and Insight We classify existing domain image generation methods for SFUDA into three subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' We present the challenges of methods in each subcatetory and our insights below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Among the above-mentioned three subcategories, the first one (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', batch normalization statistics transfer) explicitly performs BN statistics matching between source and tar- get domains for style transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Since the BN statistics of the source model are off-the-shelf, these methods are generally efficient and don’t require complex model train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' However, BN statistics mainly focus on keeping the style features while the content information cannot be well preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Therefore, this strategy is more applicable to scenarios where the contextual structure of images between source and target domains does not differ too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' It may not show good adaptation performance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', from a natural image to a cartoon image, since the content information has significant changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Note that BN statistics transfer can also be used as a pre-processing step in source-free domain adaptation, and it can be combined with other strategies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', circular learning [28], for more effective knowledge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Methods in the second subcategory (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', surrogate source data construction) aim to approximate the proxy source domain using appropriate target samples directly, fol- lowed by conventional unsupervised domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Their application is quite broad, including semantic seg- mentation [31], object recognition [30], [32], [33], image classification [29], and digital recognition [29], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In general, methods in this group are straightforward and computation-efficient by avoiding introducing extra hy- perparameters, which is different from generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' However, because the proxy source samples are directly selected from the target domain, these generated source data may not effectively represent the original source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moreover, how to effectively select informative target data for source data approximation is an important topic to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Some studies have proposed var- ious strategies for target data selection based on entropy measurement [31], source prototype [30], [32], aggregated source decision boundary [29], and equality constrained optimization [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' This is still an open but very interesting future direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For multi-source settings, it is promising to study which source predictor(s) we should refer to for effective target data selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Methods in the third category (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', GAN-based image gen- eration) typically synthesize images based on a generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Since the generator can model underlying complex distribution of source data with given random noise, GAN-based models generally create more diverse images compared with methods in second category (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', surro- gate source data construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' However, these methods introduce additional frameworks and learnable parame- ters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', generators and discriminators), which may cost more computation resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' By comparing experimental results, we find the surrogate source data construction methods [32], [33] generally outperform the GAN-based generators [36], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The possible reason may be that the constructed source data in the former are closer to real data distributions, while those recovered in GAN-based methods usually suffer from a mode collapse problem [30] that leads to low-quality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Note that the mode col- lapse problem can be partly mitigated by using a carefully tuned learning rate, manifold-guided training [47], and virtual mapping [48], which is worth exploring further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Different from image generation methods (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1) that directly generate source/target-like images, the dis- tribution generation methods (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2) generate fea- ture prototype/distribution to achieve cross-domain feature alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' By comparing the reported experimental results, we find that the distribution generation approaches [39]– [41] usually outperform the GAN-based image generation method [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' And surrogate source data construction meth- ods [30], [32] usually show superior performance compared with the distribution generation methods [39], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The underlying reason could be that the source distributions directly derived from the existing target data [30], [32] are more accurate and stable than the approximated ones [39], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' How to drive the approximated source distribution to the real one can be further explored in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2 Model Fine-Tuning Method Instead of generating source-like data for standard unsuper- vised domain adaptation, many studies attempt to fine-tune a pre-trained source model by exploiting unlabeled target data in a self-supervised training scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Based on differ- ent strategies for fine-tuning the source model, we divide existing studies into five subcategories: (1) self-supervised knowledge distillation, (2) domain alignment via statistics, (3) contrastive learning, (4) uncertainty-guided adaptation, and (5) hidden structure mining methods, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' More details are introduced in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1 Self-Supervised Knowledge Distillation Many studies [22], [49]–[55] transfer knowledge learned from source data to the target model via knowledge distil- lation in a self-supervised manner, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In these works, most of them [22], [49]–[52] achieve source-free domain adaptation via a mean-teacher scheme for knowledge transfer [56], where the target model not only learns from unseen target domain but also well preserves source model X TΦ SLCESOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY 5 Aug-α Aug-β Teacher Network Student Network EMA LKD Source Model Initialize Target Data Initialize Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Illustration of Self-Supervised Knowledge Distillation methods for source-free unsupervised domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' With target data from different augmentations as inputs, a teacher-student framework is uti- lized to exploit target features, where parameters of teacher network are usually exponential moving average (EMA) of those of student network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Aug-α and Aug-β denote two data augmentation methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', flip, rotation, shift, noise addition, distortion, etc), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' LKD: Knowl- edge distillation loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For instance, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [49] propose a self- supervised distillation scheme for automatic polyp detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' By means of keeping output consistency of weak and strong augmented polyp images, source knowledge is im- plicitly transferred to the target model with a mean teacher strategy [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Besides, a diversification flow paradigm is designed to gradually eliminate the style sensitivity among different domains, further enhancing model robustness to- wards style diversification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [50] also propose a self-supervised mean-teacher approach for knowledge distillation, with a Transformer module [57] embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' This effectively helps the target model focus on object re- gions rather than less informative background in an image, thus improving model generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Assuming that both source and target images are generated from a domain- invariant space by adding noise perturbations on each spe- cific domain, Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [51] establish a super target domain via augmenting perturbations based on the original target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The super and the original target domains are fed into a mean-teacher framework, with three consistency regularization terms (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' image, instance, and class-wise) introduced for domain alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [22] first divide the target data into clean and noisy subsets guided by a computation loss and regard them as labeled and unlabeled examples, and then utilize the mean teacher technique to self-generate pseudo-labels for the unlabeled target data for domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Instead of utilizing the conventional one-teacher one- student paradigm, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [52] construct a multi-teacher multi-student framework, where each teacher/student net- work is initialized using a public network pre-trained on a single dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Here, a graph is constructed to model the similarity among samples, and such relationship predicted by the teacher networks is used to supervise the student net- works via a mean-teacher technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Rather than leverage the mean-teacher paradigm that averages student’s weights, Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [53] propose to distill knowledge from teacher to student networks by style and structure regularizations, as well as physical prior constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Instead of employing a teacher-student network as the studies mentioned above, Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [54] achieve data-free adaptation through gradual knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Specifically, they first generate pseudo- labels via a constructed neighborhood geometry, and then Target Data Source Model Stored Source Statistics Target Model Derived Target Statistics Statistics Discrepancy Minimization Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Illustration of Domain Alignment via Statistics methods for source-free unsupervised domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The corresponding meth- ods leverage batch statistics stored in the pre-trained source model to approximate the distribution of inaccessible source data, and then perform cross-domain adaptation by reducing distribution discrepancy between source and target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' use pseudo-labels obtained from the latest epoch to super- vise the current training epoch for knowledge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2 Domain Alignment via Statistics Many studies [58]–[64] leverage batch statistics stored in the pre-trained source model to approximate the distribution of inaccessible source data, and then perform cross-domain adaptation by reducing distribution discrepancy between source and target domains, as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For example, Ishii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [58] approximate feature distribution of inaccessible source data by using batch normalization statistics (mean and variance) saved in the pre-trained source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Then, Kullback-Leibler (KL) divergence is utilized to minimize the distributional discrepancy between source and target domains, thus achieving domain-level alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Inspired by [65], [66], Paul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [60] update the mean and variance of BatchNorm [67] or InstanceNorm [68] of the pre-trained model based on unseen target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Not limited to matching low-order batch-wise statistics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', mean and variance), Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [59] additionally incorporate high-order batch-wise statistics, such as scale and shift parameters, to explicitly keep cross-domain consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moreover, they quantify each channel’s transferability based on its inter- domain divergence and assume that the channels with lower divergence contribute more to domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [61] propose to align domain statistics adaptively by modulating a learnable blending factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' By minimizing the total objective function, each BN layer can dynamically ob- tain its own optimal factor, which controls the contribution of each domain to BN statistics estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The methods mentioned above are all based on Gaussian-based statistics domain alignment, while Eastwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [62] attempt to align histogram-based statistics of the marginal feature dis- tributions of the target domain with those stored in the pre-trained source model, thus well extending adaptation to non-Gaussian distribution scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='3 Contrastive Learning Many contrastive learning studies [19], [24], [69]–[72] per- form data-free adaptation, which helps the target model capture discriminative representations among unlabeled target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The main idea is to pull instances of similar categories closer and push instances of different categories away in feature space, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' X TΦ SAug-QAug-βL KDΦ SX TSOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY 6 Before Adaptation After Adaptation Pull Close Push Apart Target Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Illustration of Contrastive Learning methods for source-free un- supervised domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' These methods exploit discriminative representations among unlabeled target data by pulling instances of similar categories closer and pushing instances of different categories away in feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For instance, Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [69] first adaptively divide tar- get instances into source-similar and source-dissimilar sets, and then design a class-aware contrastive module for cross- set distribution alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The idea is to enforce the com- pactness of target instances from the same category and reduce cross-domain discrepancy, thus prompting effective knowledge transfer from the source model to target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [70] present a cross-domain contrastive learning paradigm, which aims to minimize the distance between an anchor instance from one domain and instances from other domains that share the same category as the anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Due to the unavailability of source data, they utilize source proto- typical representations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', weight vectors in the classifier layer of a pre-trained source model, for feature alignment across two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [19] tackle the data-free domain adaptation by taking advantage of the historical source hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Specifically, they propose a historical con- trastive instance discrimination strategy to learn from target samples by contrasting their learned embeddings generated by the currently adapted and historical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' And they also design a historical contrastive category discrimination strategy to weight pseudo-labels of target data to learn category-discriminative target representations, by calculat- ing the consistency between the current and historical model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The two discrimination strategies help exploit historical source knowledge, bypassing the dependence on source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Inspired by [73], Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [71] introduce a pair-wise contrastive objective function to reduce intra- category distance and meanwhile increase inter-category distance based on generated target pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' They also introduce robust source and target models by taking advan- tage of the generated adversarial instances, which facilitates robust transfer of source knowledge to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='4 Uncertainty-Guided Adaptation Uncertainty can measure how well the target model fits the data distribution [74], and many studies [75]–[82] utilize such valuable information to guide target predictions in source-free adaptation scenarios (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For instance, Fleuret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [75] estimate uncertainty based on differences between predicted outputs with and without Dropout operation [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' By minimizing such differences, the prediction uncertainty on target data is reduced, meanwhile the learnable feature abstractor can be more robust to noise perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [76] exploit aleatoric uncertainty by encouraging intra-domain consistency between target images and their augmented ones and enforcing inter-domain feature Target Data XT Target Model Uncertainty Measurement Updated e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', Monte Carlo Dropout, Entropy, Confidence, Consistency Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Illustration of Uncertainty-Guided Adaptation methods for source- free unsupervised domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' These studies utilize uncertainty to guide target predictions, and such valuable information can be mea- sured by Monte Carlo Dropout, entropy, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' distribution consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [77] introduce a predic- tion denoising approach for a cross-domain segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In this study, a key component is introducing pixel- wise denoising via uncertainty evaluation using Monte Carlo Dropout [84], [85], which calculates the standard deviation of several stochastic outputs and keeps it under a manually- designed threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In this way, the noisy pseudo-labels can be filtered out, helping improve pseudo-label quality to achieve effective adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [78] also propose an uncertainty-guided pseudo-labeling denoising scheme, but they use soft label correction instead of manually discarding unreliable data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Specifically, they first identify misla- beled data points by utilizing a joint distribution matrix [86], [87], and then assign larger confident weights to those with higher certainty based on Monte Carlo Dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Combining target data and the corresponding rectified pseudo-labeling, a commonly used cross-entropy objective function can be leveraged for training the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Sharing the similar idea, Hegde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [79] allocate lower weights for uncer- tain pseudo-labels, where the uncertainty is measured by prediction variance based on Monte Carlo Dropout [84], [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Considering that using Monte Carlo Dropout [84] for uncertainty estimation requires manual hyperparameter adjustment [88], Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [80] quantify source model’s un- certainty using a Laplace approximation [89], [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For model training, they assign smaller weights to those target samples that are farther away from source hypothesis (measured by uncertainty), avoiding misalignment of dissimilar samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Pei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [81] tackle the uncertainty issue from the perspec- tive of improving source model transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Specifically, they estimate channel-aware transferability of the source model to target data based on an uncertainty distance, which measures the closeness between target instances and source distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' With the aim of dynamically exploiting the source model and target data, the target model obtains the source knowledge from the transferable channels and neglects those less-transferable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Unlike previous stud- ies, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [82] quantify uncertainty using self-entropy and propose a self-entropy descent mechanism to seek the optimal confidence threshold for robust pseudo-labeling of target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' They also leverage false negative mining and mosaic augmentation [91] to further eliminate the negative influ- ence of noisy labels to enhance adaptation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='5 Hidden Structure Mining Many studies [20], [92]–[98] take into consideration intrinsic feature structures of target domain and update the target model via clustering-aware pseudo-labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 10, we illustrate the main idea of hidden structure mining methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY 7 Before Adaptation Class Centroid Iteratively Update Clustering Centroid After Adaptation Target Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Illustration of Hidden Structure Mining methods for source-free unsupervised domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' These methods take into consider- ation intrinsic feature structures of target domain and iterate between target model refinement and clustering centroid update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For example, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [20] observe that target data can intrinsically form a certain cluster structure that can be used for domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Specifically, they estimate affinity among target data by taking into account the neighborhood patterns captured from local-, reciprocal-, and expanded- neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Source-free adaptation is achieved by encour- aging consistent predictions for those with high affinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Similarly, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [92] also exploit neighborhood struc- ture information of target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' They propose a local struc- ture clustering strategy to encourage prediction consistency among k-nearest target features, thus pushing target data with semantically similar neighbors closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [93] leverage semantic constraints hidden in geometric structure among target data to encourage robust clustering based on a cognition mechanism [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Source hypothesis transfer (SHOT) [94] and SHOT++ [95] attempt to mine the feature structure of the target domain, but they cannot fully ex- ploit the meaningful context since the used self-supervised pseudo-labeling does not take into account each dimen- sion’s covariance in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' To address this issue, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [96] utilize GMM in the target domain to obtain data structure, and design a joint model-data structure score to concurrently exploit source and target knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [97] propose a novel neighborhood structure cluster- ing method, which encourages intra-cluster target features closer and meanwhile disperses those inter-cluster target predictions far away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [98] utilize neighbor structure information from a new aspect by proposing a generic and model smoothness-assisted Jacobian norm regularization term, which is used to manipulate the consistency between each target instance and its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' This Jacobian norm regularizer can be easily plugged into existing source-free domain adaptation frameworks for boosting performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Different from the above mentioned methods, some studies tackle source-free domain adaptation from other perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [100] achieve data-free adaptation from an adversarial-attack aspect, which aims to generate adversar- ial target instances by adding diverse perturbations to attack the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Then, mutual information maximization is performed between representations extracted by the source and target model for the same target instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The above two steps are performed alternatively, by which the domain- invariant source knowledge can be preserved and the rich target patterns can be well explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Instead of explor- ing domain-invariant features for cross-domain knowledge transfer, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [101] mine domain-invariant parameters stored in the source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' They assume that only partial domain-invariant parameters of the source model contribute to domain adaptation, and their goal is to capture such pa- rameters while penalizing the domain-specific ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [102] explore source-free adaptation from the perspec- tive of minimum centroid shift, with the aim of searching a subspace where target prototypes are mildly shifted from source prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' An alternating optimization scheme is leveraged for model convergence and target pseudo-label update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Inspired by maximum classifier discrepancy [14], Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [103] introduce an auxiliary bait classifier for cross- domain feature alignment combined with the source anchor classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' These two classifiers aim to collaboratively push uncertain target representations to the correct side of the source classifier boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='6 Challenges and Insight We classify existing model fine-tuning methods for SFUDA into five subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The challenges of methods in each subcatetory and our insights are presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The methods in the first subcategory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', self-supervised knowledge distillation, interpret source-free domain adap- tation as a knowledge extraction and transfer process, aiming to learn domain-invariant feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Most exiting studies transfer source knowledge to the target model via a mean teacher strategy [56], where teacher weights are an exponential moving average of student weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Hence, model parameters of both teacher and student networks are tightly coupled, which may lead to a performance bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' A possible solution is to introduce a dual-student framework and let one student learn features flexibly, which may disentangle teacher- student weights to some extent [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The second subcategory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', domain alignment via statistics, leverages batch statistics stored in a pre-trained source model to approximate distribution of inaccessible source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Compared with other categories, these statistics- based methods are lightweight and prone to generalize to other tasks, since they require only a few update steps of batch-wise statistics parameters and are potentially appli- cable to real-time deployment [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' However, they are not suitable for problems that use deep network architectures without batch normalization layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The methods in the third subcategory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', contrastive learn- ing, aim to bring similar-class samples closer and push dissimilar-class samples apart based on generated tar- get pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Therefore, if the pseudo-labels contain much noise, these methods may suffer from substantial performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moreover, a memory bank is usually required to store the similarity relationship be- tween current and historical feature representations of target data, which could bring memory burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' It is interesting to investigate the storage- and transmission- efficient contrastive learning strategies in source-free set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In addition, several recent studies [105], [106] have shown that data pair construction is crucial for effective contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' One solution is utilizing contrastive information between target data and their augmented ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Previous studies [107] often use either strong or weak transformations for data augmentation, where strong augmentations mostly distort the structures of original images (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', shape distortion) while weak augmentations usually limit transformations to preserve the images’ structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', flip).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Here we propose to SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY 8 dynamically mix strong and weak augmentation of target data, which may help learn more robust representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The methods in fourth subcategory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', uncertainty-guided adaptation, focus on reducing prediction uncertainty of tar- get data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Many studies [77], [78] use Monte Carlo Dropout for uncertainty estimation, but this technique requires specialized network architecture design and model train- ing, bringing troublesome hyperparameter tuning [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' A recent study [77] points out that their method can only handle problems with minor domain shift, and performs poorly on problems with severe domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' It is inter- esting to explore this challenging problem in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The last subcategory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', hidden structure mining, considers intrinsic clustering structure of the target domain, assum- ing that geometric structure of target data may provide informative context [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The advantage of these methods is that no auxiliary frameworks are required, and thus, they can be easily incorporated into other adaptation frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' However, these methods have at least three disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' (1) Most existing studies need to iterate between feature clustering and model update, which may hinder training efficiency and cause a memory burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' (2) These methods may be infeasible to handle extremely large-scale datasets due to the difficulty of saving global latent feature embeddings of the whole dataset [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' (3) Most studies construct target geometric structures in Euclidean space, which may not be suitable for problems with non-Euclidean data such as graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Thus, how to improve training efficiency and deal with the large-size dataset, as well as mining geometry information of non- Euclidean data deserve further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' From the application perspective, computation-efficient approaches are more applicable for pixel-wise semantic segmentation tasks, which require higher resources com- pared with classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' And those memory-intensive approaches such as contrastive learning may be not suitable for semantic segmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moreover, it is worth noting that data generation methods detailed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1 can be used in conjunction with the model fine-tuning methods described in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For instance, one can first generate a virtual source domain by selecting appropriate target samples, and thus a standard unsupervised domain adap- tation framework could be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' To further exploit target information, we then take account of geometric structure of target samples and generate corresponding target pseudo- labels to fine-tune the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' These two steps can be optimized iteratively, helping generate more representative source domain and refine the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 3 BLACK-BOX SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION Different from white-box methods, in the setting of black-box source-free domain adaptation, both the source data {XS, YS} and detailed parameters of the source model ΦS are not accessi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Only the hard or soft model predictions of the target data XT from the source model ΦS are leveraged for do- main adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Depending on the utilization of the black- box predictor, the existing black-box SFUDA studies can be mainly divided into three categories: Self-Supervised Knowledge Distillation, Pseudo-Label Denoising, and Generative Distribution Alignment methods, with details introduced below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1 Self-Supervised Knowledge Distillation Some studies [109]–[114] construct a teacher-student-style network architecture with knowledge distillation to trans- late the source knowledge to the target domain in a self- supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For instance, Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [109], [110] enforce output consistency between a source model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', teacher) and a customized target model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', student) via a self-distillation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Specifically, a memory bank is first constructed to store the prediction of each target sample based on the black-box source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' This source model then acts as a teacher to maintain an exponential moving averaging of source and target prediction following [115], [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Additionally, structural regularization on the target domain is further incorporated during adaptation for more effective knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Similarly, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [111], [112] employ an exponential mixup decay scheme to explicitly keep prediction consistency of source and target domains, thus gradually capturing target-specific feature representa- tions and obtaining the target pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [113] extend the teacher-student paradigm from image analysis to more challenging video analysis, where not only spa- tial features but also temporal information are taken into consideration during domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For knowledge distillation, the target model is regarded as a student, which aims to learn similar predictions generated by a teacher (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', source) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The teacher model is meanwhile updated to maintain an exponential moving averaging prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Instead of distilling knowledge between source and target domains, Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [114] transfer knowledge between the target network and its subnetwork in a mutual way, where the subnetwork is a slimmer version generated from the original target network following Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' And target features are extracted by leveraging multi-resolution input images, helping improve the generalization ability of the target network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moreover, a novel data augmentation strategy, called frequency MixUp, is proposed to empha- size task-related regions-of-interests while simultaneously reducing background interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2 Pseudo-Label Denoising Some studies [118], [119] tackle domain shift by carefully denoising unreliable target pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For example, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [118] combat noisy pseudo-labels via noise rate estimation, which first preserves more training samples at the start of the training process following [120] and then gradually filters out the noisy ones based on their loss values as training proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The pseudo-labels are itera- tively refined according to a category-dependent sampling strategy, encouraging the model to capture more diverse representations to improve model generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Dif- ferent from Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [118] that only select part of reliable target data during model training, Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [119] take into account all target data and rectify noisy pseudo-labels from a negative learning aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Specifically, their approach assigns complementary ground-truth labels for each target sample, helping alleviate error accumulation for noisy pre- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moreover, a maximum squares objective function is SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY 9 utilized as confidence regularization to prevent the target model from being trapped in easy sample training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [121] incorporate pseudo-label denoising and self-supervised knowledge distillation into a unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Specifically, domain knowledge is first distilled from the trained source predictor to warm up the target model by an exponential moving averaging scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The unlabeled target domain is then split into two subsets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', easy and hard groups) according to their adaptation difficulty [122], and the Mix- Match strategy [123] is leveraged to progressively exploit all target representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In this way, the noise accumulation is further suppressed, thereby improving the efficacy of pseudo-label denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='3 Generative Distribution Alignment Different from the above methods, some studies perform distribution alignment across domains in a generative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For instance, Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [124] perform domain adaptation by maximizing the lower bound in variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Specifically, they construct a generation path as well as an inference path, where the generation path produces a prior feature distribution derived from predicted category labels, and the inference path approximates a posterior feature distribution based on each target instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The latent dis- tribution alignment can be achieved by maximizing the evi- dence lower vound in variational inference for cross-domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Similarly, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [125] also construct the generation and inference paths, but they achieve adaptation via minimizing the upper bound of the prediction error of target data in variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [126] achieve source-free adaptation by first building multiple source models and then generating a virtual intermediate surrogate domain to select target samples with minimum inconsistency predicted by the source models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The knowledge transfer is achieved by feature distribution alignment between the virtual surrogate domain and the target domain based on a joint probability maximum mean discrepancy [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='4 Challenges and Insight In this section, we classify existing black-box SFUDA meth- ods into three categories based on how they utilize the noisy target predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The challenges of each category and our insights are presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The first category, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', self-supervised knowledge distillation, aims to gradually transfer source knowledge to a cus- tomized target model by enforcing output consistency between a teacher (source) and a student (target) network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' This learning strategy has also been used in white-box SFUDA (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The difference is that model weights of student networks are accessible in white-box SFUDA methods, but not in black-box SFUDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In black- box SFUDA, instead of leveraging any parameter details, the teacher network is only updated by source predic- tions and historical target predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The two items are typically weighted by a momentum factor, which helps dynamically adjust their contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Self-supervised knowledge distillation has shown promising performance in object recognition [109], semantic segmentation [111], and video action recognition tasks [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The methods in the second category, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', pseudo-Label denoising, tackle black-box SFUDA from the perspective of noisy label rectification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' It has shown that a pseudo-label denoising approach [118] has inferior performance than the self-supervised knowledge distillation method [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The reason may be that the former [118] only focuses on noisy prediction itself while neglecting target data struc- ture that is well considered in the latter [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Considering that pseudo-label denoising methods can tackle unbal- anced label noise via noise rate estimation, combining pseudo-label denoising with self-supervised knowledge distillation strategies will be a promising future direc- tion, especially in class-imbalance scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moreover, if the black-box predictor only provides one-hot hard predictions instead of probability predictions, the utility of methods in this subcategory will be greatly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The reason is that the noise rate cannot be well estimated in practice, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', there is nearly no difference between the output of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='45, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='55] and that of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='95] because the source predictor produces the same output (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The third category, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', generative distribution alignment, at- tempts to perform domain adaptation by minimizing fea- ture distribution discrepancy across the source and target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Since source distribution is inaccessible in black- box models, some generative approaches are utilized to generate such reference distribution for target data to align with, including variational autoencoder [124], [125] and surrogate source domain construction [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' These methods are more suitbale for recognition/classification tasks, but less suitable for semantic segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For example, generating surrogate feature distribution of an object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', car) is usually easier than that of a semantic scene (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', cityscape), since the latter contains different objects and thus the pixel-wise neighborhood relationship is difficult to model in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Besides the strategies proposed above, it is also crucial to build a general and robust black-box source model, with which the target predictions tend to be more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' To achieve that, one possible solution is augmenting the diversity of source data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', adding some perturbation) before constructing the source model, which may eliminate style discrepancy between two domains, thus improving the generalization ability of the source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Another solution is using soft probability labels instead of hard one-hot labels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', [0, 1]) for model training, which prevents the source model from being over-confident and helps enhance its generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Compared to white-box methods, there are relatively few black-box SFUDA methods as well as benchmark datasets, which needs to be further explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 4 DISCUSSION In this section, we first compare the white-box and black- box SFUDA methods and then summarize several useful strategies to improve model generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' We also list datasets commonly used in the field in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1 Comparison of White-Box and Black-Box SFUDA By comparing existing white-box and black-box SFUDA methods, we have the following interesting observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY 10 TABLE 1 Commonly used datasets for evaluating the performance of source-free unsupervised domain adaptation (SFUDA) approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Dataset Domain # Instance # Category # Description Digit Recognition Digits-Five [128] 5 215,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='695 10 MNIST [129],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' SVHN [130],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' USPS [131],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' MNIST-M [13],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Synthetic Digits [13] Semantic Segmentation Segmentation datasets 4 45,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='766 GTA5 [132],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Cityscapes [133],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' SYNTHIA [134],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' NTHU [135] Object Recognition Office-31 [136] 3 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='652 31 Amazon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Webcam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' DSLR Office-Home [137] 4 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='500 65 Artistic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Clip Art,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Product,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Real-World VisDA [138] 2 280,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='000 12 Synthetic and real images Office-Caltech-10 [139] 4 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='533 10 Amazon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' DSLR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Webcam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Caltech10 ImageCLEF-DA [140] 4 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='400 12 Caltech-256 [141],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' ImageNet ILSVRC2012 [16],' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='130 Replay-Attack [150],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' OULU-NPU [151],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' CASIA-MFSD [152],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' MSU-MFSD [153] LiDAR Detection LiDAR datasets 3 158,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='510 Waymo [154],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' KITTI [155],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' nuScenes [156] Video Action Recognition UCF-HMDBfull [157] 2 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='209 12 UCF101 [158],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' HMDB51 [159] Sports-DA [160] 3 40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='718 23 UCF10 [158],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Sports-1M [161],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Kinetics [162] Daily-DA [160] 4 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='949 8 ARID [163],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' HMDB51 [159],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moments-in-Time [164],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Kinetics [162] Traffic Sign Recognition Sign datasets 2 151,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='839 43 Syn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='Signs [165],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' GTSRB [166] Image Classification VLCS [167] 4 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='729 5 Caltech101 [168],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' LabelMe [169],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' SUN09 [170],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' VOC2007 [142] Medical Data BraTS2018 [171] 2 285 Cross-disease (high and low grade glioma),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' cross-modality (T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' T1ce,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' FLAIR) MMWHS [172] 2 40 Cross-modality (magnetic resonance imaging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' computed tomography) Brain skull stripping [173] 3 35 NFBS [174],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' ADNI [175],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' dHCP [176] Polyp segmentation 4 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='718 CVC-ClincDB [177],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Abnormal Symptoms [178],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' ETIS-Larib [179],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' EndoScene [180] EEG MI Classification [126] 4 528 2/4 MI2-2 [181],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' MI2-4 [181],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' MI2015 [182],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' AlexMI [183] Prostate segmentation 2 682 NCI-ISBI 2013 Challenge [184],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' PROMISE12 challenge [185] Optic disc&cup segmentation 3 660 REFUGE [186],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' RIMONE-r3 [187],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Drishti-GS [188] Autism diagnosis 4 411 2 NYU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' USM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' UCLA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' and UM of ABIDE dataset [189] Compared with black-box SFUDA that cannot access any source parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' white-box SFUDA is capable of mining more source knowledge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', batch statistics) that facili- tates more effective domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' White-box SFUDA methods may suffer from data pri- vacy leakage problems [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For instance, Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [190] reveal that raw data can be recovered based on source image distribution via a deep inversion technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Using a membership inference attack strategy [191], [192], it is possible to infer whether a given sample exists or not in training dataset, thereby revealing private information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The black-box SFUDA can help protect data privacy be- cause only application programming interface (API) is accessible while detailed model weights are withheld, but it may suffer from performance degradation of cross- domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Most white-box SFUDA methods assume that model ar- chitecture is shared between source and target domains, while the black-box SFUDA methods try to design task- specific target models for knowledge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Such flex- ible model design in black-box SFUDA methods is very useful for target users with low computation resources, since they can design more efficient and lightweight target models for domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Black-box SFUDA methods neither require data synthe- sis nor model fine-tuning, which helps to accelerate the convergence process of model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In contrast, white- box methods are usually computationally intensive and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For instance, it is reported that the com- putational cost of a black-box SFUDA method [126] is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='83s while that of two competing white-box methods are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='17s [94] and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='43s [193], respectively, reflecting the computation efficiency of black-box SFUDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In summary, when using white-box and black-box SFUDA methods, we have to make a trade-off between obtaining better performance, protecting confidential infor- mation, and reducing computational and memory costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2 Useful Strategies for Improved Generalizability To facilitate research practice in this field, we summarize several useful techniques that could be used to improve the generalizability of learning models for source-free unsuper- vised domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1 Entropy Minimization Loss Most SFUDA methods utilize an entropy minimization loss [194] to reduce uncertainty of model predictions [27], [59], [75], [94], [111], [112], [195]–[200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' This simple yet effective strategy encourages the model to generate one-hot predictions for more confident learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2 Diversity Enforcing Loss To prevent predicted labels from collapsing to categories with larger number of samples, many studies leverage a diversity enforcing loss to encourage diverse predictions over target domain [80], [94], [196], [201]–[205] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The usual practice is to maximize the entropy of empirical label distri- bution over the batch-wise average of model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='3 Label Smoothing Technique In source-free adaptation studies, a pre-trained source model is generally obtained via training on labeled source data before adaptation stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Currently, many studies use a label smoothing technique [206], [207] to produce a robust source model [20], [30], [94], [101], [208], [209].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' This tech- nique aims to transform original training labels from hard labels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', 1) to soft labels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='95), which prevents the source model from being over-confident, helping enhance its generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Also, the experiments have shown that label smoothing can encourage closer representations of training samples from the same category [206].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' With a more general and robust source model, it is likely to boost adaptation performance on target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='4 Model Regularization Many regularization terms are utilized in existing SFUDA methods by incorporating some prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For in- stance, an early learning regularization [39], [120], [210] is used to prevent the model from over-fitting to label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' A stability regularization [38], [211]–[213] is leveraged to pre- vent parameters of the target model to deviate from those of the source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' A local smoothness regularization [38], [214] is used to encourage output consistency between the target model and its noise-perturbed counterpart, helping improve robustness of the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' A mixup regular- ization [30], [109], [114], [215], [216] is used to enforce pre- diction consistency between original and augmented data, which can mitigate the negative influence of noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='5 Confidence Thresholding Many studies leverage pseudo-labeling to train the tar- get model in a self-supervised way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Instead of utilizing a manually-designed threshold to identify reliable/confident pseudo-labels, a commonly used strategy is automatically learning the confidence threshold for reliable pseudo-label selection [217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' To further tackle the class-imbalance prob- lem, some studies [75], [78], [212], [218], [219] propose to learn dynamic threshold for each category, which provides a fair chance for categories with limited samples to generate pseudo-labels for self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 5 FUTURE OUTLOOK 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='1 Multi-Source/Target Domain Adaptation To utilize diverse and rich information of multiple domains, a few studies [29], [193], [204], [220], [221] propose multi- source data-free adaptation to transfer source knowledge to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [29] introduce a sample transport learning method, but the proposed model is shal- low, and thus cannot handle highly nonlinear feature ex- traction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' To tackle this problem, several deep learning based models [193], [204] are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' But they ignore the fact that the generated target pseudo-labels may be noisy, which may cause training bias when matching target domains with large domain gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The key to solving problems with multi- source domains is quantifying the transferability of different source models and utilizing their complementary informa- tion for promoting cross-domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Even several strategies are proposed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', aggregation weight [193] and source-specific transferable perception [204]), more explo- rations are encouraged to address the problem of negative transfer during cross-domain knowledge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' A few studies [222]–[226] incorporate federated learning into domain adaptation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Federated learning [227]– [229] is a decentralized scheme to facilitate collaborative learning among multiple distributed clients without shar- ing training data or model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The constraint that prevents data and parameter transmission across different source domains is not required in multi-source-free domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For instance, federated adversarial domain adapta- tion (FADA) introduced by Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [222] is among the first attempts to propose the concept of federated domain adaptation, which employs a dynamic attention mecha- nism to transfer knowledge from multi-source domains to an unlabeled target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' In this method, each source model needs to synchronize with the target domain after each training batch, resulting in huge computation costs and potential risk of privacy leakage [230].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' To tackle this problem, Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [223] introduces a consensus focus schema that greatly improves communication efficiency for decentralized domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moreover, Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [225] utilize a homomorphic encryption approach for privacy pro- tection, and Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [226] introduce a flexible uncertainty- aware strategy for reliable source selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' However, cur- rent federated learning studies usually produce a common model for all clients without considering heterogeneity of data distribution of different clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Therefore, the common model cannot adapt to each client adaptively, which may affect adaptation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' It would be very interesting to investigate personalized federated learning [231], with which current or new clients can easily adapt to their own local dataset by performing a few optimization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Besides, all the methods mentioned above require labeled data from multiple sources to train a federated model, in- evitably increasing annotation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Therefore, approaches that effectively exploit unlabeled data from multiple source domains in a decentralized way are urgently needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' On the other hand, Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [232] and Shenaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [233] have proposed several federated multi-target domain adap- tation strategies for transferring knowledge of a labeled source server to multiple unlabeled target clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' More ad- vanced techniques for federated multi-target domain adap- tation are highly desirable, by considering computation and communication cost, annotation burden, and privacy protection of different target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='2 Test-Time Domain Adaptation Most SFUDA approaches require pre-collected unlabeled target data for model training, termed “training-time adap- tation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Test-time adaptation [198], [234]–[237] has been investigated by adapting the source model to the target SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY 12 domain during inference procedure only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The advantages of test-time adaptation are mainly twofold: (1) The adaptation process does not need iterative training, which greatly im- proves computational efficiency, so the model can be easily deployed in an online manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' (2) Without relying on target training data, test-time adaptation is expected to be well generalized to diverse target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Even current studies have made promising achievements, there are still some problems worth exploring, listed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Some studies [198], [238], [239] need to access batch- sized (>1) target samples during inference, which can- not handle scenarios where target samples arrive one-by- one sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Two studies [240], [241] perform image- wise adaptation rather than batch-wise adaptation, but they cannot deal with cases with large distribution shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' It is interesting to explore how to handle the scenarios where test instances come from continuous changeable domains in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Additionally, the solutions on how to adap- tively exploit test data can be further explored [242], such as adjusting model weights dynamically based on sample discrepancy across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='3 Open/Partial/Universal-Set Domain Adaptation This survey focuses on close-set source-free domain adapta- tion, where the label space of source and target domains is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' But practical scenarios are much more compli- cated when the category shift issue occurs across different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' There are three non-close-set scenarios: (1) open-set (Cs ⊂ Ct) problems, (2) partial-set (Cs ⊃ Ct) problems, and (3) universal-set (Cs\\Ct ̸= ∅ and Ct\\Cs ̸= ∅, Cs ⊂ Ct, Cs ⊃ Ct) problems, where Cs and Ct denote the category label set for source and target domains, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Currently, only a few studies [18], [243], [244] attempt to handle the category shift problem in source-free adapta- tion scenarios, including out-of-distribution data construc- tion [18], [243], neighborhood clustering learning [245], uncertainty-based progressive learning [246], and mutual information maximization [203].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The main idea behind these studies is to recognize out-of-source distribution samples and improve generalization ability of the source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' However, the model performance of existing studies is not quite satisfactory due to the inaccessibility of valuable category-gap knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' One possible solution is to adap- tively learn a threshold instead of using a fixed one to determine the acceptance/rejection of each target sample as a “known” category via some similarity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moreover, some strategies used in non-source-free domain adaptation can also be borrowed, such as distribution weighted combining rule [247], category-invariant represen- tation learning [248], one-vs-all learning scheme [249], and global-local optimization [250].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='4 Flexible Target Model Design For black-box SFUDA methods, due to unavailability of structure and parameters of target model, one usually has to manually design a target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For instance, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' [111] choose a U-Net based framework as the target model for segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' However, such manually designed architectures may be not suitable when adapting to the tar- get domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' It is expected that the automatic design of target models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', using neural architecture search (NAS) [251]– [253], helps improve the learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Consider- ing that NAS has recently become a popular strategy for searching proper network architectures in deep learning, we can integrate it into SFUDA scenarios to find more proper and efficient target models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' And how to balance the search space and search cost of network parameters can be further investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moreover, hyperparameters used in NAS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', optimizer strategy, weight decay regularization) should be carefully considered since they also have a significant im- pact on network performance [254].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='5 Cross-Modality Domain Adaptation Existing studies mainly focus on one single modality for domain adaptation, while a few studies perform cross- modality adaptation in source-free settings [28], [255].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' For instance, for medical data analysis, the acquisition expense of computed tomography (CT) scans is generally less than that of magnetic resonance imaging (MRI) scans, hence it may greatly reduce annotation cost for a segmentation task when transferring a source model trained on CT images to MRI scans [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Moreover, in computer vision field, it would be promising to investigate cross-modality adaptation in the future, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', image→video, which aims to achieve video recognition based on the source model trained on the image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Also, how to effectively integrate multi-modality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', image, sound, text, and video) data for domain adapta- tion in a source-free way is an interesting but not yet widely studied problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='6 Continual/Lifelong Domain Adaptation Most current studies focus on improving adaptation per- formance on the target domain while neglecting the perfor- mance on source domain, running the risk of catastrophic forgetting problems [256].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' To address this issue, several solutions have been developed from different aspects, such as domain expansion [257], historical contrastive learn- ing [19], domain attention regularization [92], and model perturbation [258], while there is still massive room for performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Inspired by continual/lifelong learning [259]–[262], continual domain adaptation has re- cently made great progress by investigating gradient reg- ularization [263], iterative neuron restoration [264], buffer sample mixture [265], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The continual domain adaptation in source-free settings for mitigation of catastrophic forget- ting remains an underdeveloped topic that can be further explored in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='7 Semi-Supervised Domain Adaptation Source-free domain adaptation in semi-supervised settings (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', with a few labeled target data involved for model train- ing) has also been explored in recent years [197], [266], [267].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' It usually performs semi-supervised adaptation with the help of active learning [268], [269], model memorization rev- elation [270], and consistency and diversity learning [271].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' There is still a lot of space for improvement with a limited number of labeled target samples, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=', by fine-tuning the current source-free adaptation frameworks, but this is not the focus of this survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' SOURCE-FREE UNSUPERVISED DOMAIN ADAPTATION: A SURVEY 13 6 CONCLUSION In this paper, we provide a comprehensive review of recent progress in source-free unsupervised domain adaptation (SFUDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' We classify existing SFUDA studies into white- box and black-box groups, and each group is further catego- rized based on different learning strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' The challenges of methods in each category and our insights are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' We then compare white-box and black-box SFUDA meth- ods, discuss effective techniques for improving adaptation performance, and summarize commonly used datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' We finally discuss promising future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' It is worth noting that the research topic of source-free unsu- pervised domain adaptation is still in its early stages, and we hope this survey can spark new ideas and attract more researchers to advance this high-impact research field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' ACKNOWLEDGMENT This work was supported by NIH grants RF1AG073297 and R01MH108560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Voulodimos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Doulamis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} +page_content=' Doulamis, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfbffL/content/2301.00265v1.pdf'} 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100644 index 0000000000000000000000000000000000000000..cff5f8943296eca064e46e2c9da0d1bfe01675bf --- /dev/null +++ b/2NAyT4oBgHgl3EQfovg1/content/tmp_files/2301.00511v1.pdf.txt @@ -0,0 +1,2985 @@ +Asteria-Pro: Enhancing Deep-Learning Based Binary Code Similarity Detection +by Incorporating Domain Knowledge +SHOUGUO YANG, CHAOPENG DONG∗, YANG XIAO†, YIRAN CHENG, ZHIQIANG SHI‡, ZHI +LI, and LIMIN SUN, Institute of Information Engineering, Chinese Academy of Sciences, China and School of +Cyber Security, University of Chinese Academy of Sciences, China +The widespread code reuse allows vulnerabilities to proliferate among a vast variety of firmware. There is an urgent need to detect +these vulnerable code effectively and efficiently. By measuring code similarities, AI-based binary code similarity detection is applied to +detecting vulnerable code at scale. Existing studies have proposed various function features to capture the commonality for similarity +detection. Nevertheless, the significant code syntactic variability induced by the diversity of IoT hardware architectures diminishes the +accuracy of binary code similarity detection. In our earlier study and the tool Asteria, we adopt a Tree-LSTM network to summarize +function semantics as function commonality and the evaluation result indicates an advanced performance. However, it still has utility +concerns due to excessive time costs and inadequate precision while searching for large-scale firmware bugs. +To this end, we propose a novel deep learning enhancement architecture by incorporating domain knowledge-based pre-filtration +and re-ranking modules, and we develop a prototype based on Asteria called Asteria-Pro. Pre-filtration module seeks to eliminates +dissimilar functions to boost subsequent deep learning model calculations, while re-ranking module aims to raises the rankings of +vulnerable functions among candidates generated by deep learning model. Our evaluation indicates that pre-filtration module cuts +the calculation time by 96.9% and re-ranking improves MRR and Recall by 23.71% and 36.4%. By incorporating the pre-filtration and +re-ranking modules, Asteria-Pro outperforms existing state-of-the-art approaches in bug search task, by a significant large margin. +We conduct a large-scale real-world firmware bug search and Asteria-Pro manages to detect 1,482 vulnerable functions with a high +precision 91.65%. +ACM Reference Format: +Shouguo Yang, Chaopeng Dong, Yang Xiao, YIRAN CHENG, Zhiqiang Shi, Zhi Li, and Limin Sun. 2023. Asteria-Pro: Enhancing +Deep-Learning Based Binary Code Similarity Detection by Incorporating Domain Knowledge. 1, 1 (January 2023), 32 pages. https: +//doi.org/10.1145/nnnnnnn.nnnnnnn +1 +INTRODUCTION +Code reuse is very popular in IoT firmware to facilitate its development [62]. Unfortunately, code reuse also introduces +the vulnerabilities concealed in original code to numerous firmware [21]. The security and privacy of our lives +are seriously threatened by the widespread use of these firmware [63]. Even though the vulnerabilities have been +∗ First Author and Second Author contribute equally to this work. +† Corresponding author. +‡ Corresponding author. +Authors’ address: Shouguo Yang, yangshouguo@iie.ac.cn; Chaopeng Dong, dongchaopeng@iie.ac.cn; Yang Xiao, xiaoyang@iie.ac.cn; YIRAN CHENG, +chengyiran@iie.ac.cn; Zhiqiang Shi, shizhiqiang@iie.ac.cn; Zhi Li, lizhi@iie.ac.cn; Limin Sun, sunlimin@iie.ac.cn, Institute of Information Engineering, +Chinese Academy of Sciences, Beijing, China and School of Cyber Security, University of Chinese Academy of Sciences, Beijing, China. +Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not +made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components +of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to +redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. +© 2023 Association for Computing Machinery. +Manuscript submitted to ACM +Manuscript submitted to ACM +1 +arXiv:2301.00511v1 [cs.SE] 2 Jan 2023 + +2 +Shouguo Yang, Chaopeng Dong, and Yang Xiao, et al. +publicly disclosed, there are a large number of firmware that still contain them due to delayed code upgrades or code +compatibility issues [17]. Such recurring vulnerabilities, often known as N-day vulnerabilities, cannot be identified by +symbol information such as function name because such symbol information is typically stripped during the firmware +compilation. Besides, the source code of firmware is not available since IoT vendor only provides binary version of +firmware. +To this end, binary code similarity detection (BCSD) is applied to quickly finding homologous vulnerabilities in a large +amount of firmware [22]. The BCSD technique focuses on determining the similarity between two binary code pieces. +As to the vulnerability search, the BCSD focuses on finding other homologous vulnerable functions given a known +vulnerability function. In addition to the vulnerability search, BCSD has been widely used for other security applications +such as code plagiarism detection [15, 47, 56], malware detection [41, 42], and patch analysis [27, 33, 61]. Despite many +existing research efforts, the diversity of IoT hardware architectures and software platforms poses challenges to BCSD +for IoT firmware. There are many different instruction set architectures (ISA) such as ARM, PowerPC, X64, and X86 for +IoT firmware. The instructions are different and the rules, such as the calling convention and the stack layout, also +differ across different ISAs [54]. It is non-trivial to find homologous vulnerable functions across platforms. +BCSD methods can be generally classified into two categories: i) dynamic analysis-based methods and ii) static +analysis-based methods. The methods based on dynamic analysis capture the runtime behavior as function semantic +features by running target functions, where the function features can be I/O pairs of function [54] or system calls during +the program execution [28], etc. They are not scalable for large-scale firmware analysis since running firmware requires +specific devices and emulating firmware is also difficult [19, 34, 70]. The methods based on static analysis mainly +extract function features from assembly code. An intuitive way is to calculate the edit distance between assembly code +sequences [23]. They cannot be directly applied in cross-architecture since instructions are different across architectures. +Architecture-independent statistical features of functions are proposed for the similarity detection [30]. These features +are less affected across architectures such as the number of function calls, strings, and constants. Furthermore, the +control flow graph (CFG) at the assembly code level is utilized by conducting a graph isomorphism comparison for +improving the similarity detection [30, 32]. Based on statistical features and CFG, Gemini [64] leverages the graph +embedding network to encode functions to vectors for similarity detection. With the application of deep learning +models in programming language analysis, various methods have recently appeared to employ such models to encode +binary functions in different forms and calculate function similarity based on function encoding [45, 49, 53, 60]. Static +analysis-based methods are faster and more scalable for large-scale firmware analysis but often produce false positives +due to the lack of semantic information. Since homologous vulnerable functions in different architectures usually share +the same semantics, it is desirable that a cross-architecture BCSD can capture the function semantic information in a +scalable manner. +In our previous work Asteria [68], we first utilize Tree-LSTM network to encode the AST in an effort to fit its semantic +representation. In particular, Tree-LSTM is trained using a siamese [37] architecture to understand the semantic +representation by feeding homologous and non-homologous function pairs into Tree-LSTM network. As a result, +the Tree-LSTM network learns function semantic representations to distinguish homologous and non-homologous +functions. To further improve the accuracy, we also use the call graph to calibrate the AST similarity. Specifically, +we count callee functions of target functions in call graph to measure the function call difference. Final function +similarity is determined by calibrating AST similarity with the function call difference. In our previous evaluation, +Asteria outperforms the available state-of-the-art methods, Gemini and Diaphora in terms of accuracy. The evaluation +results demonstrate the superiority of function semantic extraction by encoding AST with Tree-LSTM model for BCSD. +Manuscript submitted to ACM + +Asteria-Pro +3 +However, encoding AST incurs a clear temporal cost in Asteria. According to our earlier research [68], the entire AST +encoding process takes about one second. When Asteria is applied to vulnerability detection, where, given a vulnerable +function, there are massive functions to do a similarity calculation, the time cost is unacceptable. As the majority +of candidate functions are non-homologous, there is space for Asteria’s efficiency to be enhanced. In other words, +non-homologous candidate functions differ from vulnerable functions in some characteristics that we can exploit to +skip most non-homologous functions more efficiently. In addition, vulnerability detection-like evaluation is absent +from the majority of efforts [53, 64, 65], including our prior study Asteria. It is necessary to evaluate the performance of +Asteria on the vulnerability search task. Moreover, according to the result in the real world vulnerability detection [68], +Asteria suffers from high false positives, which affects its effectiveness in reality. +There are two main challenges that hinder Asteria from being practical for large-scale vulnerability detection: +• Challenge 1 (C1). It’s challenging to filter out the majority of non-homologous functions before encoding ASTs, +while retaining the homologous ones, to speed up the vulnerability-detection process. +• Challenge 2 (C2). It’s challenging to distinguish similar but non-homologous functions. Despite Asteria’s high +precision on homologous and non-homologous classification, it still yields false positives when distinguishing +functions with similar ASTs. +We design Asteria-Pro by introducing domain knowledge with two answers A1 and A2 to overcome these two +challenges. Our fundamental concept is that introducing inter-functional domain knowledge will helps Asteria-Pro +achieve greater precision combined the intra-functional semantic knowledge deep learning model learned. Asteria- +Pro consists of three module: 1) domain knowledge-based (DK-based) pre-filtration, 2) deep learning-based (DL-based) +similarity detection, and 3) DK-based re-ranking, among them DL-based similarity detection is basically based on Asteria. +Domain knowledge is fully exploited for different purpose in DK-based pre-filtration and re-ranking. In pre-filtration +module, Asteria-Pro aims to skip as many as possible non-homologous function by comparing lightweight robust +features (A1). Meanwhile, the filtration is required to retain all homologous functions. To this end, we conduct a +preliminary study into the filtering efficacy of several lightweight function features. According findings of the study, we +propose a novel algorithm that successfully employ three distinct function features in the filter. In re-ranking module, +Asteria-Pro confirms the homology of functions by comparing call relationships (A2), based on the assumption that +functions designed for distinct purposes have different call relationships. +Our evaluation indicates that Asteria-Pro significantly outperforms existing state-of-the-art methods in terms of +both accuracy and efficiency. Compared with Asteria, Asteria-Pro successfully cuts the detection time of Asteria- +ProAsteria by 96.90% by incorporating DK-based pre-filtration module,. By incorporating DK-based re-ranking, +Asteria-Pro manages to enhance the MRR and Recall@Top-1 by 23.71% and 36.4%, to 90.8% and 89.6%, respectively. +Asteria-Pro identifies 1,482 vulnerable functions with a high precision of 91.65% by conducting a large-scale real- +world firmware vulnerability detection utilizing 90 CVEs. Moreover, the detection results of CVE-2017-13001 demonstrate +that Asteria-Pro has an advanced capacity to detect inlined vulnerable code. +Our contributions are summarized as follows: +• We conduct a preliminary study to demonstrate the effectiveness of various simple function features in identifying +non-homologous functions. +• To the best of our knowledge, it is the first work to propose incorporating domain knowledge before and after +deep learning models for vulnerability detection optimization. We implement the domain knowledge-based +pre-filtration and re-ranking algorithms, and equip them on our previous work. +Manuscript submitted to ACM + +4 +Shouguo Yang, Chaopeng Dong, and Yang Xiao, et al. +• The evaluation indicates the pre-filtration significantly reduces the detection time and re-ranking improves the +detection precision by a fairly amount. The Asteria-Pro outperforms existing state-of-the-art methods in +terms of both accuracy and efficiency. In evaluation 8.5, we find that the performance of distinct BCSD method +may vary widely in different usage scenarios. +• We demonstrate the utility of Asteria-Pro by conduct a large-scale real-world firmware vulnerability detection. +Asteria-Pro manages to find 1,482 vulnerable functions with a high precision of 91.65%. We analyze the +vulnerability distribution in widely-used software of various IoT vendors to illustrates inspiring findings. +2 +BACKGROUND +We first briefly describe the AST structure adopted in this work, followed by demonstration of the AST holding more +stable structure than CFG across architectures. Then we introduce the Tree-LSTM model utilized in AST encoding. +Finally, the broad problem definition for the application of BCSD to bug search is given. +2.1 +Abstract Syntax Tree +Table 1. Statements and Expressions in ASTs. We count the statements and expressions for nodes in ASTs after the decompilation by +IDA Pro and list the common statements and expressions. This table can be extended if new statements or expressions are introduced. +Node Type +Label +Note +Statement +if +1 +if statement +block +2 +instructions executed sequentially +for +3 +for loop statement +while +4 +while loop statement +switch +5 +switch statement +return +6 +return statement +goto +7 +unconditional jump +continue +8 +continue statement in a loop +break +9 +break statement in a loop +Expression +asgs +10∼17 +assignments, including assignment, assignment after or, xor, and, add, sub, mul, +div +cmps +18∼23 +comparisons including equal, not equal, greater than, less than, greater than or +equal to, and less than or equal to. +ariths +24∼34 +arithmetic operations including or, xor, addition, subtraction, multiplication, +division, not, post-increase, post-decrease, pre-increase, and pre-decrease +other +34∼43 +others including indexing, variable, number, function call, string, asm, and so +on. +block +if +le +block +block +var num +asg +asg +var num var var +return +call +num +void +histsizesetfn(UNUSED(p), long v) +{ + if (v< 1) +histsiz = 1; + else + histsiz = v; + resizehistents(); +} +block +if +le +block +block +var num +asg +asg +var num var var +return +call +num +ROOT +2 +7 +35 +36 +1 +12 +2 +35 +35 +7 +35 +21 +35 +35 +6 +20 +35 +35 +AST Converting +Fig. 1. Source code of function histsizesetfn and the corresponding decompiled AST of x86 architecture. +Manuscript submitted to ACM + +Asteria-Pro +5 +2.1.1 +AST Description. An AST is a tree representation of the abstract syntactic structure of code in the compilation +and decompilation process. Different subtrees in an AST correspond to different code scopes in the source code. Figure +1 shows a decompiled AST corresponding to the source code of function histsizesetfn in zsh v5.6.2 in the left. +The zsh is a popular shell software designed for interactive use, and the function histsizesetfn sets the value of +a parameter. The lines connecting the source code and AST in Figure 1 show that a node in the AST corresponds +to an expression or a statement in the source code. A variable or a constant value is represented by a leaf node in +AST. We group nodes in an AST into two categories: i) statement nodes and ii) expression nodes according to their +functionalities shown in Table 1. Statement nodes control the function execution flow while expression nodes perform +various calculations. Statement nodes include if, for, while, return, break and so on. Expression nodes include common +arithmetic operations and bit operations. +block +if +le +block +block +var num +asg +asg +var num var var +return +call +num +void +histsizesetfn(UNUSED(p), long v) +{ + if (v< 1) +histsiz = 1; + else + histsiz = v; + resizehistents(); +} +block +if +gt +block +block +var num +asg +asg +var var var num +return +call +num +(a) AST for x86 platform +(b) AST for ARM platform +sub +esp, 0ch +mov +eax, [esp+0ch+arg_4] +test +eax, eax +jle +short loc_809F187 +mov +ds:histsiz, eax +loc_809F187: +mov +ds:histsiz, 1 +jmp +short loc_809F17E +loc_809F17E: +call +resizehistents +add +esp, 0ch +retn +LDR +R3, =histsiz +CMP +R1, #0 +MOVLER2, #1 +STRGT R1, [R3] +STRLE R2, [R3] +B +resizehistents +sub +esp, 0ch +mov +eax, [esp+0ch+arg_4] +test +eax, eax +jle +short loc_809F187 +mov +ds:histsiz, eax +loc_809F187: +mov +ds:histsiz, 1 +jmp +short loc_809F17E +loc_809F17E: +call +resizehistents +add +esp, 0ch +retn +LDR +R3, =histsiz +CMP +R1, #0 +MOVLER2, #1 +STRGT R1, [R3] +STRLE R2, [R3] +B +resizehistents +(c) CFG for x86 platform +(d) CFG for ARM platform +Fig. 2. ASTs and CFGs of the function histsizesetfn under different architectures. +2.1.2 +AST Structure Superiority. Both CFG and AST are structural representations of a function. The CFG of a function +contains the jump relationships between basic blocks that contain straight-line code sequences [38]. Though CFG has +been used for similarity measurement in BCSD [30], David et al. [23] demonstrated that CFG structures are greatly +affected by different architectures. We find AST shows better architectural stability across architectures compared with +CFG since the AST is generated from the machine independent intermediate representations which are disassembled +from assemble instructions during the decompilation process [20]. Figure 2 shows the changes of ASTs and CFGs in +x86 and ARM architectures, respectively. For the CFGs from x86 to ARM, we observe that the number of basic blocks +changes from 4 to 1, and the number of assembly instructions has changed a lot. However, the ASTs, based on higher +level intermediate representation, slightly change from x86 to ARM, where the changes are highlighted with blue +boxes. Besides, AST preserves the semantics of functionality and is thus an ideal structure for cross-platform similarity +detection. +2.2 +Tree-LSTM Model +In natural language processing, Recursive Neural Networks (RNN) are widely applied and perform better than Convo- +lutional Neural Networks [69]. RNNs take sequences of arbitrary lengths as inputs considering that a sentence can +Manuscript submitted to ACM + +6 +Shouguo Yang, Chaopeng Dong, and Yang Xiao, et al. +consist of any number of words. However, standard RNNs are not capable of handling long-term dependencies due to +the gradient vanishing and gradient exploding problems. As one of the variants of RNN, Long Short-Term Memory +(LSTM) [39] has been proposed to solve such problems. LSTM introduces a gate mechanism including the input, forget, +and output gates. The gates control the information transfer to avoid the gradient vanishing and exploding (calculation +details in Section § 6.1). Nevertheless, LSTM can only process sequence input but not structured input. Tree-LSTM is +proposed to process tree-structured inputs [58]. The calculation by Tree-LSTM model is from the bottom up. For each +non-leaf node in the tree, all information from child nodes is gathered and used for the calculation of the current node. +In sentiment classification and semantic relatedness tasks, Tree-LSTM performs better than a plain LSTM structure +network. There are two types of Tree-LSTM proposed in the work [59]: Child-Sum Tree-LSTM and Binary Tree-LSTM. +Researchers have shown that Binary Tree-LSTM performs better than Child-Sum Tree-LSTM [59]. Since the Child-Sum +Tree-LSTM does not take into account the order of child nodes, while the order of statements in AST reflects the +function semantics, we use the Binary Tree-LSTM for our AST encoding. +3 +PRELIMINARY STUDY +This study aims to assess and uncover accessible function features that are effective at identifying non-homologous +functions to guide our pre-filtration construction. To evaluate the features, we prepare the code base and incorporate a +number of metrics (§ 3.1). To uncover appropriate features, we evaluate and compare popular conventional features +found in existing remarkable works (§ 3.2). +3.1 +Evaluation Benchmark +3.1.1 +Dataset. To derive robust features, we compile a large collection of binaries from 184 open source software +(OSS), including widely used OpenSSL, FFmpeg, Binutils, etc. Since our tool aims to conduct similarity detection across +different architectures, we compile these OSS to 4 distinct architectures, X86, X64, ARM, and PowerPC. In addition, +we align the default compilation settings during compilation with real-world usage. After compilation, numerous test +binaries with "test" or "buildtest" as a prefix or suffix are generated to test the software’s functionality. These test +binaries are removed from the collection because 1) their functions are simple and comprise only a few lines of code. 2) +do not participate in the real execution of software function. After removal, the binary collection retains 1,130 binaries, +or 226 for each architecture. +We construct a large dataset containing pairs of homologous and non-homologous functions. To this end, function +names are retained by software after its compilation. In order to construct homologous function pairs, we utilize binary +functions with the same function name in the same software. Otherwise, they form non-homologous functions. For +instance, if function 𝐹 is present in the source code, compilation will generate 4 versions of binary functions for distinct +instruction set architectures: 𝐹𝑥86, 𝐹𝑥64, 𝐹𝑎𝑟𝑚, and 𝐹𝑝𝑝𝑐, respectively. These variants of functions are homologous +functions for one another. We extract 132,274 unique binary functions each architecture, for a grand total of 529,096 +binary functions. We select at random 40,111 functions from each architecture, totaling 160,444 functions. Among +these, we randomly select 1000 functions as source functions and use them to perform filtering operations on all +functions. In specifically, source functions are used to evaluate the capability of the target feature to filter away +non-homologous functions while preserving homologous ones from the 40,111 functions that have been selected. We +prepare the homologous function pairs and groups for feature evaluations. Specifically, we utilize the source function +name 𝐹 together with the library name 𝐵, since function names in different binaries might be duplicated but having +different functionalities, to combine into a function identifier 𝐹𝐵. After compilation, we will get different homologous +Manuscript submitted to ACM + +Asteria-Pro +7 +binary functions 𝐹𝐵 +𝑋86, 𝐹𝐵 +𝑋64, 𝐹𝐵 +𝐴𝑅𝑀, and 𝐹𝐵 +𝑃𝑃𝐶 for different architectures, X86, X64, ARM, and PowerPC. These four +binary versions of function form a homologous function groups. We pick a random version of function to combine with +other 3 versions of functions to form 3 homologous function pairs. We calculate different metrics for function pairs and +groups respectively. +3.1.2 +Metrics. True positive rate (TPR) and false positive rate (FPR) are utilized to evaluate the filtering capability of +various features. TPR demonstrates the feature’s capacity to retain homologous functions, while FPR demonstrates its +capacity to exclude non-homologous functions. In the subsequent filtering phase, our goal is to identify features that +can filter out non-homologous functions as effectively as possible (low FPR) while maintaining all homologous functions +(very high TPR). +We employ 𝑛 source functions to evaluate the filtering capability of diverse features. For each source function 𝐹𝐵 +𝑋 , we +construct a candidate function pool of 𝑀 randomly selected binary functions, containing three homologous functions +of 𝐹𝐵 +𝑋 . Consequently, source function 𝐹𝐵 +𝑋 is used to form 3 homologous pairs and 𝑀 × 4 − 4 non-homologous pairs. The +function pair scores are calculated based on distinct feature and scores below a threshold value 𝑇 are omitted. In the +remaining function pairs, the homologous function pair is regarded as a true positive 𝑇𝑃 while the non-homologous +function pair is regarded as a false positive 𝐹𝑃. The following equations illustrate how we calculate these three metrics +for various features: +𝑇𝑃𝑅 = +�𝑛 +𝑖=1𝑇𝑃𝑝 +𝑖 +3 × 𝑛 +(1) +𝐹𝑃𝑅 = +�𝑛 +𝑖=1 𝐹𝑃𝑖 +𝑛 × (𝑀 × 4 − 4) +(2) +(3) +3.2 +Candidate Features Evaluation +We intend to identify the most efficient and effective filter features by evaluating features proposed in existing studies. +On the basis of the evaluation results, we utilize and improve the candidate features for the filter needs. +3.2.1 +Feature Selection. We collect basic features from prior research [30, 64, 65, 68] and divide them into two categories: +CFG-family feature and AST-family feature. CFG-family features include 5 types of numeric features: the number of +instructions, arithmetic instructions, call instructions, logical instructions, transfer instructions, and 2 constant features: +string constants and numeric constants [30]. Since AST must be prepared for model encoding calculation (§ 6), we +therefore summarize three syntactic characteristics as AST-family characteristics. +• No. AST Nodes. The number of AST nodes. +• AST node Cluster. The number of different node types in AST. For example, in Figure 1, the AST node Cluster is +denoted as [𝑏𝑙𝑜𝑐𝑘 : 3,𝑖𝑓 : 1,𝑟𝑒𝑡𝑢𝑟𝑛 : 1,𝑐𝑎𝑙𝑙 : 1,𝑛𝑢𝑚 : 3,𝑏𝑙𝑜𝑐𝑘 : 2,𝑎𝑠𝑔 : 2, 𝑣𝑎𝑟 : 4,𝑙𝑒 : 1] +• AST Fuzzy Hash. We first generate node sequence by traversing the AST preorder. Then we apply the fuzzy hash +algorithm [44] to generate the fuzzy hash of AST. +3.2.2 +Feature Similarity in Filtration. The format of features divides them into two types with distinct similarity +calculations: value type and sequence type. Value type features consist of No. Instruction, No. Arithmetic, No. Logic, +No. Callee, and No. AST nodes. Sequence type features consist of Numeric Constant, String Constant, AST Node Cluster, +and AST Seq Hash. For value type features, we use the relative difference ratio (𝑅𝐷𝑅) as shown below for similarity +Manuscript submitted to ACM + +8 +Shouguo Yang, Chaopeng Dong, and Yang Xiao, et al. +calculation: +𝑅𝐷𝑅(𝑉1,𝑉2) = 1 − 𝑎𝑏𝑠(𝑉1 − 𝑉2) +𝑚𝑎𝑥(𝑉1,𝑉2) +(4) +where 𝑉1,𝑉2 are feature values. For each sequence-type feature, we first sort the feature’s items and then concatenate +them into a single sequence. Then, we employ the common sequence ratio (CSR) based on the longest common sequence +(LCS) as follows: +𝐶𝑆𝑅(𝑆1,𝑆2) = 2 × 𝐿𝐶𝑆(𝑆1,𝑆2) +𝑙𝑒𝑛(𝑆1) + 𝑙𝑒𝑛(𝑆2) +(5) +where 𝑆1,𝑆2 are feature sequences, and function 𝐿𝐶𝑆(·, ·) returns the length of the longest common sequence between +𝑆1,𝑆2. The above two equations are used for similarity calculation of various features. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +FPR +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +TPR +AST Node Cluster(auc=0.978) +No. AST Nodes(auc=0.956) +No. callee(auc=0.944) +Numeric Constant(auc=0.940) +AST Seq Fuzzy Hash(auc=0.932) +No. Instructions(auc=0.862) +String Constant(auc=0.773) +No. Logic(auc=0.687) +No. Arithmetic(auc=0.521) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Similarity Threshold +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +F-score +No. callee +AST Node Cluster +AST Seq Fuzzy Hash +No. AST Nodes +Numeric Constant +No. Instructions +String Constant +No. Logic +No. Arithmetic +0 +50 +100 +150 +200 +250 +300 +350 +No. Callee +No. AST Nodes +AST Node Cluster +AST Seq Fuzzy Hash +No. Instructions +No. Arithmetic +No. Logic +String Constant +Numeric Constant +Time/10−7s +Fig. 3. ROC Curves for Traditional Features. +Fig. 4. 𝐹𝑠𝑐𝑜𝑟𝑒 of Traditional Features. +Fig. 5. Time Costs of Similarity Calculation +for Different Features. +3.2.3 +Evaluation Results. In the evaluation, the values for 𝑛 and 𝑀 in § 3.1.2 are set to 1000 and 20, 000, respectively. As +depicted in Figure 3, TPRs and FPRs calculated for each feature under various thresholds are presented as a receiver +operating characteristic (ROC) [71] curve. Additionally, We compute the area under ROC curve (AUC), which reflects +the feature’s ability to distinguish between homologous and non-homologous functions. The AUC values of the features +extracted from AST (i.e., No. AST Nodes, AST Node Cluster, and AST Seq Fuzzy Hash) are high, as presented in the image. +However, when the TPR is high, they generate a big FPR. Figure 5 depicts the time costs associated with similarity +calculations for various features. Clearly, sequence type features require more time than value type features. Nonetheless, +their time consumption falls within an acceptable range of magnitudes. At least 105 exact calculations can be completed +every second. +We observe in Figure 3 that at high TPR (0.996), the No. Callee feature produces relatively lower FPR (0.111). Recalling +the objective of the filtering phase, we aim to select features with a low FPR at a very high TPR. Features with high +AUC do not necessarily meet the our objective. For example, the feature AST Node Cluster has a higher FPR (0.47) than +feature No. Callee (FPR = 0.111) under the same TPR (0.996), even feature AST Node Cluster has higher AUC (0.978) than +feature No. Callee (AUC = 0.944). In this regard, we propose a new metric 𝐹𝑠𝑐𝑜𝑟𝑒, which indicating high TPR and FPR. +𝐹𝑠𝑐𝑜𝑟𝑒 = +1 +1 +𝑇𝑃𝑅 + 𝐹𝑃𝑅 +(6) +For each feature, we calculate the 𝐹𝑠𝑐𝑜𝑟𝑒 and plot the 𝐹𝑠𝑐𝑜𝑟𝑒 curves in Figure 4. The Figure 4 plots 𝐹𝑠𝑐𝑜𝑟𝑒 curves of +various features at various similarity thresholds. We can see that the No. callee has the best 𝐹𝑠𝑐𝑜𝑟𝑒 (0.90) at threshold +Manuscript submitted to ACM + +Asteria-Pro +9 +0.94. In this case, we summary the development challenges of No. callee and the improvement in § 5 by conducting +manually analysis. +Prefiltration +Feature Extraction +Filtering +AST +Extraction +Target +Function +Candidate +Functions +Target +Function +Remainder +Functions +Candidate +Homologous +Functions +Similarity Calculation +TreeLSTM Encoding +Siamese Calculation +Homology +Confirmation +Re-ranking +Input + DK-based Prefiltration +§ + DL-based Similarity Calculation +§ +Homologous +Functions + DK-based Re-ranking +§ +Output +Fig. 6. Workflow of Asteria-Pro. DK stands for Domain Knowledge. DL stands for Deep Learning. +4 +METHODOLOGY OVERVIEW +Asteria-Pro consists of three primary modules, DK-based Prefiltration, DL-based Similarity Calculation, and +DK-based Re-ranking as shown in Figure 6, where DK stands for Domain Knowledge, and DL stands short for Deep +Learning. DK-based prefiltration module utilizes syntactic features to filter out dissimilar functions from candidate +functions in a lightweight and efficient manner (see § 5). DL-based similarity calculation module encodes ASTs into +representation vectors using the Tree-LSTM model, and determines similarity score between target function and +remainder functions using a Siamese network (see § 6). DK-based re-ranking module reorders candidate homologous +functions in the above module using lightweight structural features (i.e., function call relationship). Asteria-Pro can +ultimately detect homologous functions across architectures efficiently and effectively. +5 +DK-BASED PREFILTRATION +At this stage, Asteria-Pro intends to incorporate an efficient and effective filter. Callee functions, which are useful in +eliminating non-homologous functions demonstrated in preliminary study, are applied extensively for this purpose. We +present a variant of the callee function as well as a novel algorithm for constructing the filter. +5.1 +Callee Exploitation Challenges +Note that the No. of callee only counts the number of callee functions but omits crucial information such as function +names. Observing that a portion of callee function names are retained after binary stripping suggests that callee +relationship has potential for additional exploration. To fully exploit the callee relationship, we manually analyze the +incorrect detection cases of No. of callee evaluations in § 3.2 and summarize the challenges to properly exploiting it: +• Exploitation Challenge 1 (EC1). Removed or decorated callee function name. For safety and size reasons, the +binary usually is stripped after compilation. The call functions whose names are removed after strip from symbol +table, can not provide assistance in a straightforward manner. In addition, there are several ways to decorate +function names by compiler [9]. The function name will differ from its original name in the source code after +decoration. +• Exploitation Challenge 2 (EC2). There is no callee functions in some functions (i.e., leaf node in call graph). +• Exploitation Challenge 3 (EC3). Function calls in binary target functions might not always be consistent with +source code. Function calls may be added or deleted due to compiler optimization. The reasons for the function call +change are function inline, intrinsic function replacement, instruction replacement for optimization, that behave +differently in different architectures. We describe this challenge in detail in Function Call Optimization. +Manuscript submitted to ACM + +10 +Shouguo Yang, Chaopeng Dong, and Yang Xiao, et al. +To overcome exploitation challenges, we design a new feature genealogist derived from callee relationship. The +novel algorithm UpRelation is proposed to exploit the new feature. +5.2 +Filter Feature Design +Our new feature intends to extract comprehensive information from the call relationship of software binary, including +the symbol information. Such call relationship can be presented by call graph. The call graph 𝐶𝐺 can be defined by giving +all functions as nodes and the call relationships between functions as edges:𝐶𝐺 = (V, E), where V = {𝑣|𝑣 is a function} +denotes the node collection and E = {(𝑢, 𝑣)|𝑢 calls 𝑣} denotes the edge collection. For edge (𝑢, 𝑣) ∈ 𝐸, we say that +function 𝑣 is a callee function of function 𝑢. The genealogist is a function list containing partial callee functions of the +target function. Considering EC1, we can not utilize all symbol information of callee functions, since callee functions +might be in removable symbol table (i.e., static symbol table), and the function name will removed. Fortunately, to +link against dynamic libraries, function names in the dynamic symbol table 𝐷𝑆𝑇 (i.e., import and export table) will +be preserved [36]. For example, if target function calls external function ‘strcpy’, the callee function name ‘strcpy’ +remains in import table rather than removed after binary strip. We define genealogist 𝐺𝐿 of target function 𝑓 as: +𝐺𝐿𝑓 = {𝑣|𝑣 ∈ V, (𝑓 , 𝑣) ∈ E, 𝑣 ∈ 𝐷𝑆𝑇 }. +For cases where a function calls the same function multiple times, we keep multiple identical function names. +5.3 +Filtration algorithm +To support our new feature, we propose a feature similarity algorithm, called UpRelation. The algorithm utilizes +context information in the call graph to overcome challenge EC2, EC3. Specifically, the algorithm utilizes parent nodes +of leaf nodes in call graph to match leaf nodes to address the EC2. In the algorithm, we adopt a drill down strategy, that +combines three feature 𝐺𝐿, No. Callee, and String Constants according to their information content, considering the +fairly robust performance of No. Callee and String Constants in preliminary evaluation. The No. Callee of function 𝑓 is +denoted by 𝐶𝑎𝑙𝑙𝑒𝑒𝑓 and the String Constant set of function 𝑓 is denoted by 𝑆𝑡𝑟𝐶𝑜𝑛𝑠𝑓 . +Given a vulnerable function 𝑓𝑣, Algorithm 1 aims to omit the most of non-homologous functions, with retaining +vulnerable candidate function to a list (𝑉 𝐹𝐿) from target function list 𝑇𝐹𝐿. Code from line 2 to line 6 performs filtering +when feature genealogist 𝐺𝐿𝑓 𝑣 of 𝑓𝑣 is not empty. Specifically, the algorithm calculates the 𝐶𝑆𝑅 between 𝐺𝐿𝑓 𝑣 and +𝐺𝐿𝑓 of all candidate functions in line 4. Then it filters out functions whose 𝐶𝑆𝑅 is less than a threshold 𝑇𝐺𝐿. Similarly, +when callee number 𝐶𝑎𝑙𝑙𝑒𝑒𝑓 𝑣 of 𝑓 𝑣 is not 0, it filters out functions by calculating 𝑅𝐷𝑅 score from line 7 to line 11. +Line 12 to 19 makes up the most crucial portion of the algorithm, which matches the leaf functions to address EC2. +All caller functions of 𝑓 𝑣 are first visited in this section of the algorithm at line 14. It employs 𝑈𝑝𝑅𝑒𝑙𝑎𝑡𝑖𝑜𝑛 to discover +all functions that are similar to caller function, 𝑓 𝑝. For each similar function 𝑓 𝑝′, the algorithm regards all its callee +functions as vulnerable candidate functions at line 16. We find the same leaf functions by locating the same caller +functions because the caller functions of the same leaf functions are also the same. However, this introduces some +extraneous (leaf) functions, that share the same caller function but are not the same as the leaf function. We utilize +string similarity at line 22, to remove extraneous functions. After filtering by callees and strings, the algorithm finally +gets the expected vulnerable candidate function list 𝑉 𝐹𝐿. +6 +DL-BASED SIMILARITY CALCULATION +This module calculates the similarity between two function ASTs by encoding them into vectors and applying the +Siamese architecture to calculate similarity between encoded vectors. Figure 7 depicts the calculation flow. +Manuscript submitted to ACM + +Asteria-Pro +11 +Algorithm 1: UpRelation +Input: Vulnerable Function 𝑓 𝑣, Target Function List 𝑇𝐹𝐿, Thresholds 𝑇𝐺𝐿,𝑇𝑐𝑎𝑙𝑙𝑒𝑒,𝑇𝑠𝑡𝑟𝑖𝑛𝑔 +Output: Vulnerable Candidate Function List 𝑉 𝐹𝐿 +1 𝑉 𝐹𝐿 ← 𝑇𝐹𝐿; +2 if 𝐺𝐿𝑓 𝑣 is not null then +3 +for 𝑓 ∈ 𝑇𝐹𝐿 do +4 +𝑠 = CSR(𝐺𝐿𝑓 𝑣, 𝐺𝐿𝑓 ); +5 +if 𝑠 < 𝑇𝐺𝐿 then 𝑉 𝐹𝐿.pop(𝑓 ); +6 +end +7 else if 𝐶𝑎𝑙𝑙𝑒𝑒𝑓 𝑣 > 0 then +8 +for 𝑓 ∈ 𝑇𝐹𝐿 do +9 +𝑠 = RDR(𝐶𝑎𝑙𝑙𝑒𝑒𝑓 𝑣, 𝐶𝑎𝑙𝑙𝑒𝑒𝑓 ); +10 +if 𝑠 < 𝑇𝑐𝑎𝑙𝑙𝑒𝑒 then 𝑉 𝐹𝐿.pop(𝑓 ); +11 +end +12 else +13 +𝐹𝐿′ = ∅; +14 +for 𝑓 𝑝 ∈ GetCallers(fv) do +15 +for 𝑓 𝑝′ ∈ 𝑈𝑝𝑅𝑒𝑙𝑎𝑡𝑖𝑜𝑛(𝑓 𝑝,𝑉 𝐹𝐿) do +16 +𝐹𝐿′.add(GetCallees(𝑓 𝑝′)); +17 +end +18 +end +19 +𝑉 𝐹𝐿 = 𝐹𝐿′; +20 if 𝑆𝑡𝑟𝐶𝑜𝑛𝑠𝑓 𝑣 is not null then +21 +for 𝑓 ∈ 𝑉 𝐹𝐿 do +22 +𝑠 = CSR(𝑆𝑡𝑟𝐶𝑜𝑛𝑠𝑓 𝑣, 𝑆𝑡𝑟𝐶𝑜𝑛𝑠𝑓 ); +23 +if 𝑠 < 𝑇𝑠𝑡𝑟𝑖𝑛𝑔 then 𝑉 𝐹𝐿.pop(𝑓 ); +24 +end +25 else +26 +return +27 end +28 𝑉 𝐹𝐿; +𝑻𝒓𝒆𝒆-𝑳𝑺𝑻𝑴 +𝑻𝒓𝒆𝒆-𝑳𝑺𝑻𝑴 +𝒄𝒂𝒕 +𝒔𝒐𝒇𝒕𝒎𝒂𝒙 +AST Similarity +| − | +𝒗𝒂𝒍𝒖𝒆𝟏 +𝒗𝒂𝒍𝒖𝒆𝟐 +𝑒! +𝑒" +𝑒# +𝑐&' +𝑐&( +ℎ&' +ℎ&( +𝑐& +ℎ& +𝒉𝒓𝒐𝒐𝒕 +Encoding Vector +Similarity Calculation +AST Encoding +Node Encoding +ℎ!" +ℎ!# +𝑐!" +𝑐!# +𝑒! +u! +𝑖! +𝑐! +ℎ! +𝑜! +𝑒! +𝑓+, +𝑓+- +U" U# +X +X +∑ +X ++ ++ ++ ++ ++ +𝜎 +𝜎 +𝜎 +𝑡𝑎𝑛ℎ +𝑡𝑎𝑛ℎ +𝜎 +𝜎 +Siamese Network +⨀ +Fig. 7. The Siamese Architecture and Tree-LSTM Encoding. +Manuscript submitted to ACM + +12 +Shouguo Yang, Chaopeng Dong, and Yang Xiao, et al. +6.1 +Tree-LSTM Encoding +Given an AST, Tree-LSTM model encodes it into a representation vector. Tree-LSTM model is firstly proposed to encode +the tree representation of a sentence and summarize the semantic information in natural language processing. Tree- +LSTM model can preserve every property of the plain LSTM gating mechanisms while processing tree-structured inputs. +The main difference between the plain LSTM and the Tree-LSTM is the way to deal with the outputs of predecessors. +The plain LSTM utilizes the output of only one predecessor in the sequence input. We utilize Tree-LSTM to integrate +the outputs of all child nodes in the AST for calculation of the current node. To facilitate the depiction of the Tree-LSTM +encoding, we assume that node 𝑣𝑘 has two child nodes 𝑣𝑙 and 𝑣𝑟 . The Tree-LSTM encoding of node 𝑣𝑘 takes three types +of inputs: node embedding 𝑒𝑘 of 𝑣𝑘, hidden states ℎ𝑘𝑙 and ℎ𝑘𝑟 , and cell states 𝑐𝑘𝑙 and 𝑐𝑘𝑟 as illustrated in Figure 7. The +node embedding 𝑒𝑘 is generated by using the pre-trained model CodeT5 to embed the node 𝑣𝑘 to a high-dimensional +representation vector. ℎ𝑘𝑙, ℎ𝑘𝑟 , 𝑐𝑘𝑙, and 𝑐𝑘𝑟 are outputs from the encoding of child nodes. During the node encoding in +Tree-LSTM, there are three gates and three states which are important in the calculation. The three gates are calculated +for filtering information to avoid gradient explosion and gradient vanishing [58]. They are input, output, and forget +gates. There are two forget gates 𝑓𝑘𝑙 and 𝑓𝑘𝑟, filtering the cell states from the left child node and right child node +separately. As shown in Node Encoding in Figure 7, the forget gates are calculated by combining ℎ𝑘𝑙, ℎ𝑘𝑟, and 𝑒𝑘. +Similar to the forget gates, the input gate, and the output gate are also calculated by combining ℎ𝑘𝑙, ℎ𝑘𝑟, and 𝑒𝑘. The +details of the three types of gates are as follows: +𝑓𝑘𝑙 = 𝜎(𝑊 𝑓 𝑒𝑘 + (𝑈 𝑓 +𝑙𝑙 ℎ𝑘𝑙 + 𝑈 𝑓 +𝑙𝑟ℎ𝑘𝑟) + 𝑏𝑓 ) +(7) +𝑓𝑘𝑟 = 𝜎(𝑊 𝑓 𝑒𝑘 + (𝑈 𝑓 +𝑟𝑙ℎ𝑘𝑙 + 𝑈 𝑓 +𝑟𝑟ℎ𝑘𝑟) + 𝑏𝑓 ) +(8) +𝑖𝑘 = 𝜎(𝑊 𝑖𝑒𝑘 + (𝑈 𝑖 +𝑙 ℎ𝑘𝑙 + 𝑈 𝑖 +𝑟ℎ𝑘𝑟) + 𝑏𝑖) +(9) +𝑜𝑘 = 𝜎(𝑊 𝑜𝑒𝑘 + (𝑈 𝑜 +𝑙 ℎ𝑘𝑙 + 𝑈 𝑜 +𝑟 ℎ𝑘𝑟) + 𝑏𝑜) +(10) +where 𝑖𝑘 and 𝑜𝑘 denote the input gate and the output gate respectively, and the symbol 𝜎 denotes the sigmoid activation +function. The weight matrix𝑊 ,𝑈 , and bias 𝑏 are different corresponding to different gates. After the gates are calculated, +there are three states 𝑢𝑘, 𝑐𝑘, and ℎ𝑘 in Tree-LSTM to store the intermediate encodings calculated based on inputs ℎ𝑘𝑙, +ℎ𝑘𝑟, and 𝑒𝑘. The cached state 𝑢𝑘 combines the information from the node embedding 𝑒𝑘 and the hidden states ℎ𝑘𝑙 +and ℎ𝑘𝑟 (Equation 11). And note that 𝑢𝑘 utilizes tanh as the activation function rather than 𝑠𝑖𝑔𝑚𝑜𝑖𝑑 for holding more +information from the inputs. The cell state 𝑐𝑘 combines the information from the cached state 𝑢𝑘 and the cell states 𝑐𝑘𝑙 +and 𝑐𝑘𝑟 filtered by forget gates (Equation 12). The hidden state ℎ𝑘 is calculated by combining the information from cell +state 𝑐𝑘 and the output gate 𝑜𝑘 (Equation 13). The three states are computed as follows: +𝑢𝑘 = 𝑡𝑎𝑛ℎ(𝑊 𝑢𝑒𝑘 + (𝑈𝑢 +𝑙 ℎ𝑘𝑙 + 𝑈𝑢 +𝑟 ℎ𝑘𝑟) + 𝑏𝑢) +(11) +𝑐𝑘 = 𝑖𝑘 ⊙ 𝑢𝑘 + (𝑐𝑘𝑙 ⊙ 𝑓𝑘𝑙 + 𝑐𝑘𝑟 ⊙ 𝑓𝑘𝑟) +(12) +ℎ𝑘 = 𝑜𝑘 ⊙ 𝑡𝑎𝑛ℎ(𝑐𝑘) +(13) +where the ⊙ means Hadamard product [40]. After the hidden state and input state are calculated, the encoding of the +current node 𝑣𝑘 is finished. The states 𝑐𝑘 and ℎ𝑘 will then be used for the encoding of 𝑣𝑘’s parent node. During the AST +encoding, Tree-LSTM encodes every node in the AST from bottom up as shown in Tree-LSTM Encoding in Figure 7. +After encoding all nodes in the AST, the hidden state of the root node is used as the encoding of the AST. +Manuscript submitted to ACM + +Asteria-Pro +13 +6.2 +Siamese Calculation +This step uses Siamese architecture that integrates two identical Tree-LSTM model to calculate similarity between +encoded vectors. The details of the Siamese architecture M(𝑇1,𝑇2) are shown in Figure 7. The Siamese architecture +consists of two identical Tree-LSTM networks that share the same parameters. In the process of similarity calculation, +the Siamese architecture first utilizes Tree-LSTM to encode ASTs into vectors. We design the Siamese architecture +with subtraction and multiplication operations to capture the relationship between the two encoding vectors. After the +operations, the two resulting vectors are concatenated into a larger vector. Then the resulting vector goes through a +layer of softmax function to generate a 2-dimensional vector. The calculation is defined as: +M(𝑇1,𝑇2) = 𝑠𝑜𝑓 𝑡𝑚𝑎𝑥(𝜎(𝑐𝑎𝑡(|N (𝑇1) − N (𝑇2)|, N (𝑇1) ⊙ N (𝑇2)) ×𝑊 ))) +(14) +where 𝑊 is a 2𝑛 × 2 matrix, the ⊙ represents Hadamard product [40], | · | denotes the operation of making an absolute +value, the function 𝑐𝑎𝑡(·) denotes the operation of concatenating vectors. The softmax function normalizes the vector +into a probability distribution. Since 𝑊 is a 2𝑛 × 2 weight matrix, the output of Siamese architecture is a 2 × 1 vector. +The format of output is [𝑑𝑖𝑠𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝑠𝑐𝑜𝑟𝑒,𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝑠𝑐𝑜𝑟𝑒], where the first value represents the dissimilarity score +and the second represents the similarity score. During the model training, the input format of Siamese architecture +is < 𝑇1,𝑇2,𝑙𝑎𝑏𝑒𝑙 >. In our work, the label vector [1, 0] means 𝑇1 and 𝑇2 are from non-homologous function pairs and +the vector [0, 1] means homologous. The resulting vector and the label vector are used for model loss and gradient +calculation. During model inference, the second value in the output vector is taken as the similarity of the two ASTs, +and the similarity of ASTs is used in re-ranking. +Name Match +High Similarity +Search +... +... +Relational +Struct +Match +Fig. 8. The Re-ranking Motivation Example. In the rectangular box with dashed line are the top K candidate homologous functions of +𝐹1 produced by the search (i.e., DL-based similarity detection). Solid line arrows indicate the function call relationship (e.g., 𝐹1 calls +𝐶𝐹1 +𝑛 ). The dotted line arrows indicate the callee function match in re-ranking. +7 +DK-BASED RE-RANKING +This module seeks to confirm the homology of top k candidate functions output by Tree-LSTM network by re-ranking +them. In the prior phase, the Tree-LSTM network infers the semantic information from AST, which is a intra-functional +feature. The knowledge gained from AST is insufficient for establishing the homology of functions. In this phase, +function call relationships are used as domain knowledge to compensate for the lack of knowledge regarding the +inter-functional features of the Tree-LSTM. To this end, we design an algorithm called Relational Structure Match. +In contrast to the callee application in the pre-filtering module, this module uses more extensive information from calle +relationships, to show degree of homology of candidate functions. +Manuscript submitted to ACM + +14 +Shouguo Yang, Chaopeng Dong, and Yang Xiao, et al. +7.1 +Motivated Example +Our algorithm is based on a conforming observation to an intuitive law: If a function 𝐹1 calls function 𝐶𝐹1, then its +homologous function 𝐹 ′ +1 will also call the homologous function 𝐶𝐹 ′ +1 of 𝐶𝐹1. As depicted in Figure 8, we have 𝐹1 calls +𝐶𝐹1, and 𝐹 ′ +1 calls 𝐶𝐹 ′ +1. Assume that the search process for 𝐹1 yields top K functions containing the target homologous +function 𝐹 ′ +1. We then employ the call relations of 𝐹1 and 𝐹 ′ +1 to conduct precise callee function match for re-ranking. In +particular, callee functions of 𝐹1 are divided into two categories, named callees 𝐶𝐹1 +𝑛 and anonymous callees 𝐶𝐹1 +𝑎 . For +named callees, their names are utilized to match callees of functions between source function 𝐹1 and candidate top K +functions. For anonymous callees, we employ DL-based similarity detection to calculate similarity between callees of +functions between source function 𝐹1 and candidate top K functions. Recall the observation, the homologous function +𝐹 ′ +1 of 𝐹1 holds the most matched callees. 𝐹 ′ +1 is re-ranked in first place after candidate functions are re-ranked based on +matched callees. +7.2 +Relational Structure Match Algorithm +This algorithm aims to rescore each candidate function by fully exploiting the call relationship of target function and +candidate functions. The relational structure refers to call relations between target function and all its callee functions +as illustrated in Section § 7.1. To match relational structure, given a source function 𝐹1, the algorithm executes one of +following two distinct operations (𝑂1 and 𝑂2) based on whether the source function has callee functions or not. +𝑂1: When 𝐹1 has callee function(s), the algorithm extracts all callee functions of 𝐹1 to build mixed callee function +set (MCFS) (described below). Based on MCFS, the algorithm detects similarities between the target functions +and candidate functions as new scores. It re-ranks all candidate functions by combining the Asteria scores +( Equation 6 ) with the newly calculated match scores. The details of MCFS and match score calculation are +described below. +𝑂2: When 𝐹 has no callee function, the algorithm removes all candidate functions which have callee function(s). +Then the left candidate function are re-ranked by their Asteria scores. +Mixed Callee Function Set. The mixed callee function set of function 𝐹 is comprised of two types of callee functions: +named callee and anonymous callee. Anonymous callee refers to a type function for which the function name has been +removed for security reasons. The other type of callee functions have their names preserved because they are imported +or exported functions and the function name is necessary for external link purpose. These callee functions are called +named callee. We denote MCFS of 𝐹 with 𝐶𝑆𝐹 = {𝐶𝐹 +𝑛1, ...,𝐶𝐹 +𝑛𝑗,𝐶𝐹 +𝑎1, ...,𝐶𝐹 +𝑎𝑗 }, where 𝐶𝐹 +𝑛𝑗 denotes named callee function +and 𝐶𝐹 +𝑎𝑗 denotes anonymous callee function. +Match Score Calculation. With MCFSs of target function 𝐹1 and all candidate functions extracted, the algorithm +conducts two kinds of matches between callees to calculate match score 𝑀 for each candidate function. +Named Callee Match. For all named callees 𝐶𝐹1 +𝑛𝑗 in 𝐶𝑆𝐹1, the algorithm matches them and named callees of each +candidate functions by function name. For example, the named callee 𝐶𝐹1 +𝑛 and 𝐶𝐹 ′ +1 +𝑛 share the same function name +in Figure 8 so they are matched. The number of matched functions of candidate function 𝐹𝑖 is denoted as N𝐹𝑖 +𝑛 . +Anonymous Callee Match. For all anonymous callee 𝐶𝐹1 +𝑎𝑖 in 𝐶𝑆𝐹1, the algorithm utilizes DL-based similarity +detection to calculate simialrity scores between all anonymous callee of all candidate functions. For each +anonymous callee 𝐶𝐹𝑖 +𝑎𝑗 in candidate function 𝐹𝑖, the algorithm takes the maximum similarity score between it +and all anonymous callees of target function as its score S𝐹𝑖 +𝑎𝑗. +Manuscript submitted to ACM + +Asteria-Pro +15 +After matching all callee functions of candidate functions, for candidate function 𝐹𝑖, the match score 𝑀𝐹𝑖 of 𝐹𝑖 is +calculated as follows: +𝑀𝐹𝑖 = N𝐹𝑖 +𝑛 + +∑︁ +S𝐹𝑖 +𝑎𝑗 +(15) +where S𝐹𝑖 +𝑎𝑗 ∈ 𝐶𝑆𝐹𝑖 . +Match Score-based Re-ranking. The re-ranking score of candidate function 𝐹𝑖 is combined by match score 𝑀𝐹𝑖 +and its DL-based similarity MF⟩ in Equation 14. We calculate new score 𝑆𝑟𝑒−𝑟𝑎𝑛𝑘 +𝐹𝑖 +for all candidate functions with +following equation: +𝑆𝑟𝑒−𝑟𝑎𝑛𝑘 +𝐹𝑖 += 𝛼 × MF⟩ + 𝛽 × 𝑀𝐹𝑖 +(16) +where 𝛼 + 𝛽 = 1. All candidate functions are resorted by their new rank scores 𝑆𝑟𝑒−𝑟𝑎𝑛𝑘 +𝐹𝑖 +in descending order. +8 +EVALUATION +We aim to conduct a comprehensive practicality evaluation of various state-of-the-art function similarity detection +methods for bug search. To this end, we adopt 8 different metrics to depict the search capability of different methods in +a more comprehensive way. Furthermore, we construct a large evaluation dataset, in a way that is closer to practical +usage of function similarity detection. +8.1 +Research Questions +In the evaluation experiments, we aim to answer following research questions: +RQ1. How does Asteria-Pro compare to baseline methods in cross-architecture function similarity detection on the +two detection tasks? +RQ2. What is the performance of Asteria-Pro, compared to baseline methods in Task-V? +RQ3. How much do DK-based filtration and DK-based re-ranking improves in accuracy and efficiency? +RQ4. How does Asteria-Pro perform in real-world bug search? +8.2 +Implementation Details +We utilize IDA Pro 7.5 [4] and its plug-in Hexray Decompiler to decompile binary code for AST extraction. This +version of Hexray Decompiler currently supports the architectures of x86, x64, PowerPC (PPC), and ARM. We use the +Hexray Decompiler to decompile the target binaries and extract the ASTs. For the encoding of leaf nodes in Formulas +(7)-(12), we assign the state vectors ℎ𝑘𝑙, ℎ𝑘𝑟 , 𝑐𝑘𝑙, and 𝑐𝑘𝑟 to zero vectors. The loss function for model training is BCELoss, +which measures the binary cross entropy between the labels and the predictions. The AdaGrad optimizer is applied +for gradient computation and weight-matrix updating after losses are computed. Since the The computation steps of +Tree-LSTM depend on the shape of the AST, therefore, we cannot perform parallel batch computation, which makes +the batch size always to be 1. The model is trained for 60 epochs. Our experiments are performed on a local server +equipped with two Intel(R) Xeon(R) CPUs E5-2620 v4 @ 2.10GHz, each with 16 cores, 128GB of RAM, and 4T of storage. +The code of Asteria-Pro runs in a Python 3.6 environment. We use gcc v5.4.0 compiler to compile source code in +our dataset, and use the buildroot-2018.11.1 [1] for the dataset construction. We use the tool binwalk [5] to unpack +the firmware for obtaining the binaries to conduct further analysis. In UpRelation algorithm of filtering module, we +set values of thresholds 𝑇𝐺𝐿,𝑇𝑐𝑎𝑙𝑙𝑒𝑒,𝑇𝑠𝑡𝑟𝑖𝑛𝑔 to 0.1, 0.8, 0.8 based on their 𝐹𝑠𝑐𝑜𝑟𝑒, respectively. The crucial threshold 𝑇𝐺𝐿 +Manuscript submitted to ACM + +16 +Shouguo Yang, Chaopeng Dong, and Yang Xiao, et al. +is discussed in § 8.7.1. We merely set 𝛼 = 0.1, 𝛽 = 0.9 in Equation 16 to emphasize role of callee function similarities in +re-ranking. +8.3 +Comprehensive Benchmark +To compare BCSD methods in a comprehensive way, we build an extensive benchmark based on multiple advanced +works [50, 60, 64]. The benchmark comprises of two datasets, two detection tasks, and six measure metrics, where two +datasets are utilized separately for each of the two detection tasks. +8.3.1 +Dataset. The functions not involved in the prefiltering test (see § 3) are divided into two datasets for model +training and testing and evaluation. +Model Dataset Construction. We select 31,940 functions from 1944 distinct binaries to construct 314,852 homologous +function pairs and 314,852 non-homologous function pairs, respectively. Then, we divided all function pairs by an 8:2 +ratio into training set and testing set. This dataset is constructed for Tree-LSTM training and testing. +Evaluation Dataset Construction. We randomly select 60,221 functions from each architecture’s 2,875 binaries. Using +these functions, two sub-datasets for two evaluation tasks are constructed. The first sub-dataset is constructed in a +classification manner for technique comparison [64, 65], and the second sub-dataset is constructed in a bug search +manner. Each dataset includes a large number of to-be-matched tuples, each of which is in form of (𝐹, 𝑃𝑠𝑒𝑡), where 𝑃𝑠𝑒𝑡 +denotes a function set containing candidate functions to be matched with source function 𝐹. Given a source function +𝐹, the first sub-dataset combines one of its homologous function 𝐹ℎ and a non-homologous function 𝐹𝑛 as 𝑃𝑠𝑒𝑡. To +emulate the bug search, the second sub-dataset puts more non-homologous functions into the 𝑃𝑠𝑒𝑡. Specifically, given a +source function 𝐹, we randomly choose a homologous function and pick 𝑁 non-homologous functions into the 𝑃𝑠𝑒𝑡. +We call the first sub-dataset g-dataset and the second sub-dataset v-dataset. In v-dataset, we randomly pick 10, 000 +non-homologous functions for the 𝑃𝑠𝑒𝑡. The formal representation of two datasets are as following: +g-dataset : {(𝐹, (𝐹ℎ, 𝐹𝑛)), } +(17) +v-dataset : {(𝐹, (𝐹ℎ, 𝐹𝑛1, ..., 𝐹𝑛𝑖, ..., 𝐹𝑛10000)), } +(18) +where 𝐹ℎ denotes the homologous functions, and 𝐹𝑛𝑖 denotes the ith non-homologous function. For each dataset, the +source function 𝐹 will be matched with all functions in the 𝑃𝑠𝑒𝑡 for evaluation. +8.3.2 +Metrics. We choose five distinct metrics for comprehensive evaluation from earlier works [53, 60, 68]. In our +evaluation, the similarity of a function pair is calculated as a score of 𝑟. Assuming the threshold is 𝛽, if the similarity +score 𝑟 of a function pair is greater than or equal to 𝛽, the function pair is regarded as a positive result, otherwise a +negative result. For a homologous pair, if its similarity score 𝑟 is greater than or equal to 𝛽, it is a true positive (TP). +If a similarity score of 𝑟 is less than 𝛽, the calculation result is a false negative (FN). For a non-homologous pair, if a +similarity score 𝑟 is greater than or equal to 𝛽, it is a false positive (FP). When the similarity score 𝑟 is less than 𝛽, it is a +true negative (TN). These metrics are described as following: +• TPR. TPR is short for true positive rate. TPR shows the accuracy of homologous function detection at threshold +𝛽. It is calculated as 𝑇𝑃𝑅 = +𝑇𝑃 +𝑇𝑃+𝐹𝑁 . +• FPR. FPR is short for false positive rate. FPR shows the accuracy of non-homologous function detection at +threshold 𝛽. It is calculated as 𝐹𝑃𝑅 = +𝐹𝑃 +𝐹𝑃+𝑇 𝑁 . +Manuscript submitted to ACM + +Asteria-Pro +17 +• AUC. AUC is short for area under the curve, where the curve is termed Receiver Operating Characteristic (ROC) +curve. The ROC curve illustrates the detection capacity of both homologous and non-homologous functions as +its discrimination threshold 𝛽 is varied. AUC is a quantitative representation of ROC. +• MRR. MRR is short for mean reciprocal rank, which is a statistic measure for evaluating the results of a sample +of queries, ordered by probability of correctness. It is commonly used in retrieval experiments. In our evaluation, +it is calculated as 𝑀𝑅𝑅 = +1 +|𝑃𝑠𝑒𝑡 | +� +𝐹ℎ𝑖 ∈𝑃𝑠𝑒𝑡 +1 +𝑅𝑎𝑛𝑘𝐹ℎ𝑖 , where 𝑅𝑎𝑛𝑘𝐹ℎ𝑖 denotes the rank of function 𝐹ℎ𝑖 in pairing +candidate set 𝑃𝑠𝑒𝑡, and |𝑃𝑠𝑒𝑡 | denotes the size of 𝑃𝑠𝑒𝑡. +• Recall@Top-k. It shows the capacity of homologous function retrieve at top k detection results. The top k +results are regarded as homologous functions (positive). It is calculated as follows: +𝑔(𝑥) = + + +1 +if 𝑥 = 𝑇𝑟𝑢𝑒 +0 +if 𝑥 = 𝐹𝑎𝑙𝑠𝑒 +𝑅𝑒𝑐𝑎𝑙𝑙@𝑘 = 1 +|𝐹 | +∑︁ +𝑔(𝑅𝑎𝑛𝑘𝑓 𝑔𝑡 +𝑖 +≤ 𝑘) +To demonstrate the reliability of the ranking results, we adopt Recall@Top-1 and Recall@Top-10. +8.3.3 +Detection Tasks. We present the following two function similarity detection tasks based on BCSD applications [35]: +• C-task. C-task stands for classification task. This task aims to test the ability of the methods to discriminate +between homologous and non-homologous functions. C-task performs binary classification for homologous and +non-homologous functions on dataset g-dataset and calculates three metrics: TPR, FPR, AUC, since they are +generally used to indicate the binary classification performance of the model. The g-dataset meets the dataset +requirement and is used in this task. +• V-task. V-task stands for bug (vulnerability) search task. This task aims to evaluate the capacity of identifying +vulnerable functions from a vast pool of candidate functions. In other words, given a source function, it is the +same action to find its homologous function from candidate functions. This data requirement is met by the +v-dataset. More specifically, for a to-be-matched tuple (𝐹, 𝑃𝑠𝑒𝑡), tested methods calculate function similarity +between source function 𝐹 and all functions in 𝑃𝑠𝑒𝑡. After similarity calculation, the functions in the 𝑃𝑠𝑒𝑡 can be +sorted by similarity scores. Metrics MRR, Recall@Top-1, and Recall@Top-10 are calculated in this task. +8.4 +Baseline Methods. +We choose various representative cross-architectural BCSD works, that make use of AST or are built around deep +learning encoding. These BCSD works consist of Diaphora [2], Gemini [64], SAFE [51], and Trex [53]. Moreover, we also +use our previous conference work Asteria as one of baseline methods. We go over these works in more details below. +Diaphora. Diaphora performs similarity detection also based on AST. Diaphora maps nodes in an AST to primes +and calculates the product of all prime numbers. Then it utilizes a difference function to calculate the similarity between +the prime products. We download the Diaphora source code from github [2], and extract Diaphora’s core algorithm for +AST similarity calculation for comparison. Noting that it would take a significant amount of time (several minutes) to +compute a pair of functions with extremely dissimilar ASTs, we add a filtering computation before the prime difference. +The filtering calculates the AST size difference and eliminates function pairs with a significant size difference. We +publish the improved Diaphora source code on our website [6]. +Manuscript submitted to ACM + +18 +Shouguo Yang, Chaopeng Dong, and Yang Xiao, et al. +Table 2. AUCs in Task-C. +Methods +X86-ARM +X86-X64 +X86-PPC +ARM-X64 +ARM-PPC +X64-PPC +Average +Asteria +0.995 +0.998 +0.998 +0.995 +0.998 +0.999 +0.997 +Gemini +0.969 +0.984 +0.984 +0.973 +0.968 +0.984 +0.977 +SAFE +0.851 +0.867 +- +0.872 +- +- +0.863 +Trex +0.794 +0.891 +- +0.861 +- +- +0.849 +Diaphora +0.389 +0.461 +0.397 +0.388 +0.455 +0.400 +0.415 +Gemini. Gemini encodes ACFGs (attributed CFGs) into vectors with a graph embedding neural network. The ACFG +is a graph structure where each node is a vector corresponding to a basic block. We have obtained Gemini’s source +code and its training dataset. Notice that in [64] authors mentioned it can be retrained for a specific task, such as the +bug search. To obtain the best accuracy of Gemini, we first use the given training dataset to train the model to achieve +the best performance. Then we re-train the model with the part of our training dataset. Gemini supports similarity +detection on X86, MIPS, and ARM architectures. +SAFE. SAFE works directly on disassembled binary functions, does not require manual feature extraction, is compu- +tationally more efficient than Gemini. In their vulnerability search task, SAFE outperforms Gemini in terms of recall. +SAFE supports three different instruction set architecture X64, X86, and ARM. We retrain SAFE based on the official +code [51] and use retrained model parameter for our test. In particular, we select all appropriate function pairs from the +training dataset, whose instruction set architectures are supported by SAFE. Then we extract the function features for +all function pairs selected and discard the function pairs whose features SAFE cannot extract. After feature extraction, +27,580 function pairs of three distinct architecture combinations (i.e., X86-X64, X86-ARM, and X64-ARM) are obtained +for training. Next, We adopt the default model parameters (e.g., embedding size) and training setting (e.g. training +epoches) to train SAFE. +Trex. Trex is based on pretrained model [53] of the state-of-the-art NLP technique, and micro-traces. It utilizes a +dynamic component to extract micro-traces and use them to pretrain a masked language model. Then it integrates +pretrained ML model into a similarity detection model along with the learned semantic knowledge from micro-traces. +It supports similarity detection of ARM, MIPS, X86, and X64. +8.5 +Comparison of Cross-architecture Similarity Detection (RQ1) +We evaluate various approaches on two distinct tasks. Due to the fact that the baseline methods may not be able to +detect for all four instruction set architectures, the detection results for some architecture combinations are empty. In +the two paragraphs that follow, the outcomes of two distinct tasks are discussed. +Comparison on task-C. For task-C, we evaluate all approaches by performing similarity detection on all supported +architectural combinations. After detection, the three task-C-referenced metrics are calculated and presented in Table 2 +and Figure 9. In each subplot of Figure 9, the x-axis represents FPR, while the y-axis represents TPR. Six subplots +depict the ROC curves of various architecture combinations. In general, methods with performance curves closer to the +upper-left corner have a superior performance. It is evident from all subplots that our model outperforms all baseline +methods. Quantitatively, the AUC values of our model are greater than those of the other baseline techniques for any +architectural combination in Table 2. +Manuscript submitted to ACM + +Asteria-Pro +19 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +FPR +TPR +X86-ARM +Asteria +Gemini +SAFE +Diaphora +Trex +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +FPR +TPR +X86-X64 +Asteria +Gemini +SAFE +Diaphora +Trex +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +FPR +TPR +X86-PPC +Asteria +Gemini +Diaphora +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +FPR +TPR +ARM-X64 +Asteria +Gemini +SAFE +Diaphora +Trex +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +FPR +TPR +ARM-PPC +Asteria +Gemini +Diaphora +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +FPR +TPR +X64-PPC +Asteria +Gemini +Diaphora +Fig. 9. ROC Curves on All Cross-architecture Combination Detection. +Table 3. MRR and Recall of Different Methods. +Metrics +Methods +X86-X64 +X86-ARM +X86-PPC +X64-ARM +X64-PPC +ARM-PPC +Avg +Asteria-Pro +0.934 +0.887 +0.931 +0.879 +0.919 +0.903 +0.908 +Asteria +0.776 +0.724 +0.731 +0.708 +0.713 +0.750 +0.734 +Trex +0.414 +0.206 +- +0.309 +- +- +0.310 +Gemini +0.478 +0.250 +0.325 +0.336 +0.357 +0.256 +0.334 +Safe +0.029 +0.007 +- +0.009 +- +- +0.015 +MRR +Diaphora +0.023 +0.019 +0.020 +0.019 +0.020 +0.021 +0.020 +Asteria-Pro +0.917 +0.868 +0.912 +0.879 +0.899 +0.903 +0.896 +Asteria +0.706 +0.648 +0.652 +0.627 +0.631 +0.675 +0.657 +Trex +0.274 +0.110 +- +0.192 +- +- +0.192 +Gemini +0.405 +0.180 +0.242 +0.261 +0.279 +0.229 +0.266 +Safe +0.004 +0.002 +- +0.002 +- +- +0.003 +Recall@Top-1 +Diaphora +0.021 +0.016 +0.017 +0.016 +0.017 +0.018 +0.018 +Asteria-Pro +0.961 +0.921 +0.962 +0.913 +0.952 +0.932 +0.940 +Asteria +0.902 +0.867 +0.882 +0.857 +0.866 +0.890 +0.877 +Trex +0.710 +0.452 +- +0.575 +- +- +0.579 +Gemini +0.615 +0.383 +0.482 +0.478 +0.502 +0.468 +0.488 +Safe +0.022 +0.010 +- +0.014 +- +- +0.015 +Recall@Top-10 +Diaphora +0.029 +0.024 +0.026 +0.026 +0.025 +0.027 +0.026 +Comparison on task-V. As shown in Table 3, we calculate MRR, Recall@Top-1, and Recall@Top-10 for a variety of +architectural combinations. Recall@Top-1 is a rigorous metric that indicates the robust detection capacity of homologous +functions, whereas Recall@Top-10 demonstrates the capability to rank homologous functions in the top ten positions. +In the first column of the table are the metrics, and in the second column are the names of the methods. The third +through eighth columns provide the metric values for the various architectural combinations, whereas the last column +displays the mean for all architectures. Asteria-Pro and Asteria consistently outperform baseline approaches by a +Manuscript submitted to ACM + +20 +Shouguo Yang, Chaopeng Dong, and Yang Xiao, et al. +1 +CK_RV proxy_C_DigestInit(...){ +2 +/* +3 +Variable Initialization. +4 +*/ +5 +v5 = (Proxy *)self [1]. +C_GetSlotInfo; +6 +v7 = handle; +7 +result = map_session_to_real( +v5 ,&v7 ,&map ,V3); +8 +if (! result) +9 +result = map.funcs ->C_ +DigestInit(v7 , +mechanism); +10 +return result; +11 +} +1 +CK_RV proxy_C_DigestKey(...){ +2 +/* +3 +Variable Initialization. +4 +*/ +5 +v5 = (Proxy *)self [1]. +C_GetSlotInfo; +6 +v7 = handle; +7 +result = map_session_to_real( +v5 ,&v7 ,&map ,V3); +8 +if (! result) +9 +result = map.funcs ->C_ +DigestKey(v7 , mechanism +); +10 +return result; +11 +} +Fig. 10. Two Proxy Functions with only distinctions highlighted in red +significant margin across all architecture configurations. Asteria-Pro achieves a very high average MRR (0.908), +indicating an MRR improvement of up to 23.71% compared to Asteria. Even after retraining, Safe’s detection results +demonstrate that it improperly recognizes small functions despite its poor performance. Regarding Recall@Top-1, +Asteria-Pro and Asteria attain relatively high average precisions (0.89 and 0.65, respectively) that are 237% and 146% +greater than the best result (0.26). Asteria-Pro has a 36.4% improvement in Recall@Top-1 vs Asteria. Regarding +the metric Recall@Top-10, Asteria-Pro and Asteria continue to reign supreme. In comparison to Recall@Top-1, we +observe that the recall of other methods, such as Trex, increases dramatically, from 0.192 to 0.579, indicating that they +are able to rate homologous sequences quite highly. However, they are still far below Asteria-Pro. +Despite the similar ROC curve performance of Asteria, Gemini, SAFE, and Trex, their Task-V performance is entirely +different. Gemini, for instance, has a large AUC score of 0.977, which is similar to Asteria’s 0.997. However, in terms of +MRR, Gemini performs poorly (0.333) compared to Asteria (0.734) and Asteria-Pro (0.908). This suggests that testing +BCSD approaches in a single experiment setting (such as the Task-C setting) is insufficient to demonstrate application +behavior. +False Positive Analysis. There are two primary causes for Asteria’s false positive outcomes. +• Cause 1. Proxy functions hold similar syntactic structures, resulting in similar semantic. Figure 10 illustrates two +proxy functions that differ solely on line 9. Asteria-Pro fails to differentiate proxy functions since their semantics +are similar. In addition, the callees are difficult to confirm due to indirect jump table when symbols are lacking. +• Cause 2. Compilers for distinct architectures utilize various intrinsic functions that substitute libc function calls +with optimized assembly instructions. For instance, the gcc-X86 compiler may replace the memcpy function with +several memory operation instructions, causing the function to be absent from the callee function list. Asteria’s +filtering and re-ranking modules lack a complete set of callee functions for score calculation, resulting in a loss +of precision. +Manuscript submitted to ACM + +Asteria-Pro +21 +Answer to RQ1: Asteria-Pro demonstrates superior accuracy in both Task-C and Task-V. In Task-C, +dominant model in Asteria-Pro demonstrates the best classification performance by producing the highest +AUC (0.997). Regarding Task-V, Asteria-Pro outperforms other baseline methods by a large margin in +MRR, Recall@Top-1, and Recall@Top-10. In particular, Asteria-Pro has 172%, 236%, 147% higher MRR, +Recall@Top-1, Recall@Top-10 than the best baseline methods. Compared with Asteria, Asteria-Pro manages +to improve it for Task-V with 23.71% higher MRR, 36.4% higher Recall@Top-1, and 7.2% higher Recall@Top-10. +0 +2 +4 +6 +Asteria-Pro +Asteria +Trex +Gemini +Safe +Diaphora +Time/10−2s +(a) Average Feature Extraction Time for One Function +0 +50 +100 +150 +200 +250 +300 +Asteria-Pro +Asteria(/10) +Trex(/100) +Gemini +Safe +Diaphora(/10) +Time/s +Phase 1 +Phase 2 +(b) Average Time for One Search +Fig. 11. Performance Comparison of All Methods on Task-V +8.6 +Performance Comparison (RQ2) +In this section, the detection time of function similarity for all baseline approaches and Asteria-Pro are measured. +Since the DK-based prefiltration and DK-based re-ranking modules are intended to enhance performance in Task-V, we +only count the timings in Task-V. In task-V, given a source function, methods extract the function features of source and +all candidate functions, which is referred to as phase 1. Next, the extracted function features are subjected to feature +encoding and encoding similarity computation to determine the final similarities, which is referred to as phase 2. +As shown in Figure 11a, we calculate the average feature extraction time for each function. The x-axis depicts +extraction time, while the y-axis lists various extraction methods. During phase 1, Asteria-Pro, Asteria, and Diaphora +all execute the same operation (i.e., AST extraction), resulting in the same average extraction time. Since AST extraction +requires binary disassembly and decompilation, it requires the most time compared to other methods. Trex requires +the least amount of time for feature extraction, which is less than 0.001s per function, as code disassembly is the only +time-consuming activity. +Figure 11b illustrates the average duration of a single search procedure for various methods. The phases 1 and 2 of a +single search procedure are denoted by distinct signs. Due to its efficient filtering mechanism, Asteria-Pro requires +the least amount of time (58.593s) to complete a search. Due to its extensive pre-training model encoding computation, +Trex is the most time-consuming algorithm. Asteria-Pro cuts search time by 96.90%, or 1831.36 seconds, compared +to Asteria (1889.96 seconds). +Manuscript submitted to ACM + +22 +Shouguo Yang, Chaopeng Dong, and Yang Xiao, et al. +Answer to RQ2: Asteria-Pro costs the least average time to accomplish task-V. Compared with Asteria, +Asteria-Pro cuts search time by 96.90% by introducing the filtering module. +Table 4. Accuracy of Different Module Combination +Module Combination +MRR +Recall@Top-1 +Recall@Top-10 +Average Time(s) +Pre-filtering + Asteria +0.824 +0.764 +0.929 +57.8 +Asteria + Reranking +0.882 +0.864 +0.910 +1889.8 +8.7 +Ablation Experiments (RQ3) +To demonstrate the progresses made by different modules of DK-based filtration and DK-based re-ranking, we conduct +ablation experiments by evaluating the different module combinations in Asteria-Pro. The module combinations +are Pre-filtering + Asteria and Asteria + Re-ranking. The two module combinations performs Task-V and the results +are shown in Table 4. For Asteria + Re-ranking, the top 20 similarity detection results are re-ranked by the Re-ranking +module. +Filtration Improvement. Compared to Asteria, the integration of pre-filtering improves MRR, Recall@Top-1, and +Recall@Top-10 by 12.26%, 16.29%, and 5.93%, respectively. In term of efficiency, it cuts search time by 96.94%. The Pre- +filtering + Asteria combination performs better than Asteria + Re-ranking in terms of Recall@Top-10 and time consumption. +It generates a greater Recall@Top-10 because it filters out a large proportion of highly rated non-homologous functions. +Re-ranking Improvement. Compared to Asteria, the integration of Re-ranking module improves MRR, Recall@Top- +1, and Recall@Top-10 by 20.16%, 31.51%, and 3.76%, respectively. In terms of efficiency, it costs average additional 0.13s +for re-ranking, which is negligible. Compared to Pre-filtering + Asteria, re-ranking module contributes to an increase in +MRR and Recall@Top-1 by enhancing the rank of homologous functions. +Answer to RQ3: The filtering significantly cuts the calculation time by 96.94%, and increase precision slightly. +Re-ranking improves MRR, Recall@Top-1, and Recall@Top-10 by 20.16%, 31.51%, and 3.76%, respectively, with +negligible time costs. +Table 5. Capacity to Filter of Various Filtering Thresholds. +𝑇𝐺𝐿 +# Filtered Function +Recall +0.1 +9666.7 +0.9813 +0.2 +9734.1 +0.9808 +0.3 +9777.4 +0.9791 +0.4 +9793.5 +0.9773 +0.5 +9805.5 +0.9737 +Manuscript submitted to ACM + +Asteria-Pro +23 +8.7.1 +Different Filtering Threshold. In Algorithm 1, the threshold 𝑇𝐺𝐿 determines the number of functions that are +filtered out. We evaluate the efficacy of the filtering module by utilizing various𝑇𝐺𝐿 values, and the results are presented +in Table 5. The threshold values range from 0.1 to 0.5 in the first column, where a higher threshold value suggests a +more severe selection of the similarity function. The second column indicates the number of functions omitted by the +filter, while the third column displays the recall rate in the filteration results. As the threshold value increases, the recall +rate declines and the number of filtered-out functions grows. We use 0.1 as our threshold value for two key reasons: a) +the high recall rate of filtering results is advantageous for subsequent homologous function detection, and b) there is no +significant difference in the number of functions that are filtered out. +8.8 +Real World Bug Search (RQ4) +To assess the efficacy of Asteria-Pro, we conduct a massive real-world search for bugs. To accomplish this, we +obtain firmware and compile vulnerability functions to create a firmware dataset and a vulnerability dataset. Utilizing +vulnerability dataset, we then apply Asteria-Pro to detect vulnerable functions in the firmware dataset. To confirm +vulnerability in the resulting functions, we design a semi-automatic method for identifying vulnerable functions. +Through a comprehensive analysis of the results, we discover intriguing facts regarding vulnerabilities existed in IoT +firmware. +Table 6. Vulnerability Dataset +Software +CVE # +Disclosure Years +Vulnerable Version Range +OpenSSL +22 +2013∼2016 +[1.0.0, 1.0.0s] +[1.0.1, 1.0.1t] +[1.0.2, 1.0.2h] +Busybox +10 +2015∼2019 +[0.38, 1.29.3] +Dnsmasq +14 +20{15,17,20,21} +[2.42, 2.82], 2.86 +Lighttpd +10 +20{08,10,11,13,14,15,18} +[1.3.11, 1.4.49] +Tcpdump +36 +2017 +[3.5.1, 4.9.1] +8.8.1 +Dataset Construction. In contrast to our prior work, we expand both the vulnerability dataset and the firmware +dataset for a comprehensive vulnerability detection evaluation. +Vulnerability Dataset. The prior vulnerability dataset of 7 CVE functions is enlarged to 90, as shown in Table 6. Vul- +nerability information is primarily gathered from the NVD website [11]. As shown in the first column, the vulnerabilities +are collected from widely used open-source software in IoT firmware, including OpenSSL, Busybox, Dnsmasq, Lighttpd, +and Tcpdump. In the second column, the number of software vulnerabilities is listed. In the third column, the timeframe +or specific years of the disclosure of the vulnerability are listed. The final column describes the software version ranges +affected by vulnerabilities. Note that the version ranges are obtained by calculating the union of all versions mentioned +in the vulnerability reports. As a result, Asteria-Pro is expected to generate vulnerability detection results for all +software versions falling within the specified ranges. +Firmware Dataset. We download as much of firmware from six popular IoT vendors as we could, consisting of +Netgear [3], Tp-Link [14], Hikvision [10], Cisco [7], Schneider [13], and Dajiang [8] as shown in first column of Table 7. +These firmware are utilized by routers, IP cameras, switches, and drones, all of which play essential parts in our life. The +Manuscript submitted to ACM + +24 +Shouguo Yang, Chaopeng Dong, and Yang Xiao, et al. +Table 7. Firmware Dataset and Its Software Statistics. # denotes number. +Firmware Dataset +Software Statistics +Vendor +Firmware # +Binary # +Function # +OpenSSL +Busybox +Dnsmasq +Lighttpd +Tcpdump +Netgear +548 +984 +2,627,143 +349 +512 +85 +14 +24 +TP-Link +95 +177 +427,795 +66 +90 +11 +3 +7 +Hikvision +90 +92 +279,299 +55 +35 +0 +0 +2 +Cisco +29 +66 +60,396 +23 +26 +10 +5 +2 +Schneider +10 +20 +31,228 +7 +9 +2 +2 +0 +Dajiang +7 +16 +57,275 +7 +7 +1 +0 +1 +All +779 +1,355 +3,483,136 +507 +679 +109 +24 +36 +second column shows the firmware numbers, which range from 7 to 548. The third and fourth columns gives numbers +of binaries and functions after unpacking firmware by using binwalk. Note that the binary number is the number of +software selected to be in the vulnerability dataset. The fifth column to ninth column gives the five software numbers +in all firmware vendors. OpenSSL and Busybox are widely integrated in these IoT firmware as their numbers are close +to those of the firmware. Through querying their official websites for device type information, we find that the majority +of Hikvision vendor firmware is for IP cameras, whereas Cisco vendor firmware is for routers. In particular, IP camera +firmware incorporates less software than router firmware because routers offer more functionality. For example, the +firmware of the Cisco RV340 router includes OpenSSL, Tcpdump, Busybox, and Dnsmasq, whereas the majority of IP +camera firmware only include OpenSSL. Similarly, the majority of the firmware of Netgear and Tp-Link consists of +routers, while Schneider and Dajiang’firmware include specialized devices such as Ethernet Radio and Stabilizers. +8.8.2 +Large Scale Bug Search. Asteria-Pro is employed to identify vulnerable homologous functions among 3,483,136 +firmware functions by referencing 90 functions from the vulnerability dataset. Specifically, in order to expedite the +detection process, vulnerability detection is restricted to the same software between firmware dataset and vulnerability +dataset. For instance, the vulnerable functions disclosed in OpenSSL are utilized to detect vulnerable homologous +functions in OpenSSL in the firmware dataset. For each software 𝑆, we first extract features (i.e., ASTs and call graphs) of +all functions in firmware dataset and vulnerable functions in vulnerability dataset. For each vulnerability disclosed in 𝑆, +the pre-filtration module uses the call graph to filter out non-homologous functions, followed by the Tree-LSTM model +encoding all remaining functions as vectors. Asteria-Pro then computes the AST similarity between the vulnerable +function vectors and the firmawre function vector. Asteria-Pro computes reranking scores based on the top 20 of +AST similarities based on similarity scores, since the evaluation demonstrates a very high recall in the top 20. As a +final step, Asteria-Pro generates 20 candidate homologous functions for each 𝑆 as a bug search result for each +vulnerability. To further refine the bug search results, we compute the average similarity score of homologous functions +in Section 8.5 and use it to eliminate non-vulnerable functions. In particular, the average similarity score of 0.89 is used +to eliminate 3987 of 5604 results. We perform heuristic confirmation of vulnerability for the remaining 1617 results. +Vulnerability Confirmation Method. We devise a semi-automatic method for confirming the actual vulnerable +functions from the candidate homologous functions. The method makes use of the symbols and string literals within +the firmware binaries of the target. Specifically, we use unique regular expressions to match version strings for each +software and to extract function symbols from the software. The method is then comprised of two distinct operations +that correspond to two distinct vulnerable circumstances 𝑉𝐶1, 𝑉𝐶2. +Manuscript submitted to ACM + +Asteria-Pro +25 +• 𝑉𝐶1. In this circumstances, the target binary contains version string (e.g., “OpenSSL 1.0.0a”) and the symbol of +target function is not removed. +• 𝑉𝐶2. Target binary contains version strings whereas the symbol of vulnerable homologous function is removed. +The versions of software listed in Table 6 are easy to extract using version strings [26]. The descriptions of the two +confirmation operations 𝐶𝑂1 and 𝐶𝑂2 are as follows: +• 𝐶𝑂1. For 𝑉𝐶1, we confirm the vulnerable function based on the version and name of the target software. In +particular, a vulnerable function is confirmed when the following two conditions are met: 1) software version is +in vulnerable version range, 2) the vulnerable function name retains after elimination with average similarity +score. +• 𝐶𝑂2. For 𝑉𝐶2, if the software versions are in the range of vulnerable versions, we manually compare the code +between the CVE functions and remaining functions to confirm the vulnerability. +Table 8. Numbers of Vulnerable Functions, Software, Firmware in Confirmation Results. +vendor +Vulnerable Function # +Vulnerable Software # +Vulnerable +Firmware # +OpenSSL +Busybox +Dnsmasq +Lighttpd +Tcpdump +All +OpenSSL +Busybox +Dnsmasq +Lighttpd +Tcpdump +All +Netgear +367 +0 +31 +0 +26 +424 +133 +0 +7 +0 +10 +150 +145 (26.46%) +TP-Link +394 +9 +0 +2 +5 +410 +36 +3 +0 +2 +5 +46 +36 (37.89%) +Hikvision +553 +0 +0 +0 +12 +565 +52 +0 +0 +0 +1 +53 +53 (58.89%) +Cisco +0 +0 +0 +0 +2 +2 +0 +0 +0 +0 +2 +2 +2 (6.90%) +Schneider +10 +0 +0 +0 +0 +10 +1 +0 +0 +0 +0 +1 +1 (10.00%) +Dajiang +70 +0 +0 +0 +1 +71 +7 +0 +0 +0 +1 +8 +7 (100.00%) +Total +1,394 +9 +31 +2 +46 +1,482 +229 +3 +7 +2 +19 +260 +244 (31.32%) +CVE-2016-2180 +CVE-2016-0797 +CVE-2016-2105 +CVE-2015-0286 +CVE-2016-2176 +CVE-2015-1788 +CVE-2015-1790 +CVE-2015-0287 +CVE-2015-0289 +CVE-2016-2160 +1 +50 +100 +7 +7 +7 +7 +7 +7 +7 +0 +7 +7 +52 +52 +52 +52 +47 +52 +52 +52 +52 +32 +117 +27 +27 +18 +27 +16 +16 +16 +14 +12 +1 +1 +1 +1 +1 +1 +1 +0 +1 +1 +30 +30 +30 +35 +30 +29 +29 +25 +15 +29 +Number +Dajiang +Hikvision +NETGEAR +Schneider +Tplink +Fig. 12. # of Vulnerable Functions Detected from Five Vendors +Results Analysis. In Table 8, we tally the number of vulnerable functions, software, and firmware upon vulnerability +confirmation. The first column contains the names of different vendors. The second through sixth columns show the +amount of vulnerable functions in various software, while the seventh column indicates the total number of vulnerable +functions across all vendors. The eighth through twelfth columns display the amount of vulnerable software binaries in +various software, while the thirteenth column provides the total number of vulnerable software binaries. According to +the seventh column of Table 7, there are a total of 1,482 vulnerable functions. 1456 are confirmed by 𝑉𝑂1, whereas +26 are confirmed by 𝑉𝑂2. For a total of 1,456 𝑉𝑂1 vulnerable functions, 1,377 vulnerable functions rank first and 79 +Manuscript submitted to ACM + +26 +Shouguo Yang, Chaopeng Dong, and Yang Xiao, et al. +1.0.0 +1.0.0f +1.0.0g +1.0.1e +1.0.1i +1.0.1j +1.0.1m +1.0.1p +1.0.1q +1.0.1t +1.0.2c +1.0.2d +1.0.2h +CVE-2013-0166 +CVE-2014-3508 +CVE-2014-8275 +CVE-2015-0286 +CVE-2015-0287 +CVE-2015-0288 +CVE-2015-0289 +CVE-2015-0292 +CVE-2015-1788 +CVE-2015-1790 +CVE-2015-1793 +CVE-2015-3195 +CVE-2016-0797 +CVE-2016-2105 +CVE-2016-2106 +CVE-2016-2176 +CVE-2016-2180 +CVE-2016-2182 +CVE-2016-2160 +0 +0 +0 +8 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +2 +8 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +2 +8 +4 +2 +0 +0 +0 +0 +0 +0 +0 +1 +1 +2 +8 +4 +2 +0 +0 +0 +0 +0 +0 +0 +1 +1 +0 +8 +4 +2 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +6 +0 +1 +0 +0 +0 +0 +0 +0 +0 +1 +1 +2 +8 +0 +2 +0 +0 +0 +0 +0 +0 +0 +1 +1 +2 +8 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +8 +4 +2 +2 +0 +0 +0 +0 +0 +0 +0 +0 +0 +8 +4 +2 +2 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +4 +0 +0 +0 +0 +0 +0 +0 +0 +2 +1 +0 +0 +0 +0 +0 +0 +0 +0 +8 +4 +2 +2 +1 +1 +0 +4 +5 +0 +0 +0 +0 +8 +4 +2 +2 +1 +1 +0 +4 +5 +0 +0 +0 +0 +0 +0 +2 +2 +1 +1 +0 +0 +0 +0 +0 +0 +0 +8 +4 +2 +2 +1 +1 +0 +4 +5 +0 +0 +0 +0 +8 +0 +2 +2 +1 +1 +2 +4 +5 +92 +0 +0 +0 +2 +0 +2 +2 +1 +1 +10 +0 +0 +0 +0 +0 +0 +8 +4 +0 +0 +0 +0 +0 +0 +0 +0 +0 +20 +40 +60 +80 +(a) Netgear +1.0.0d +1.0.0e +1.0.0f +1.0.0g +1.0.1e +1.0.2d +CVE-2013-0166 +CVE-2014-3508 +CVE-2014-8275 +CVE-2015-0286 +CVE-2015-0287 +CVE-2015-0288 +CVE-2015-0289 +CVE-2015-0292 +CVE-2015-1788 +CVE-2015-1790 +CVE-2016-0797 +CVE-2016-2105 +CVE-2016-2176 +CVE-2016-2180 +CVE-2016-2182 +CVE-2016-2160 +0 +0 +0 +0 +29 +0 +1 +1 +3 +1 +29 +0 +1 +1 +3 +1 +29 +0 +1 +1 +3 +1 +29 +0 +0 +0 +3 +0 +22 +0 +0 +0 +0 +0 +5 +0 +1 +1 +3 +1 +9 +0 +1 +1 +3 +1 +29 +0 +0 +0 +0 +0 +29 +0 +0 +0 +0 +0 +29 +0 +0 +0 +0 +0 +29 +1 +0 +0 +0 +0 +29 +1 +0 +0 +0 +0 +29 +1 +0 +0 +0 +0 +29 +1 +0 +0 +0 +0 +2 +0 +0 +0 +0 +0 +29 +0 +0 +5 +10 +15 +20 +25 +(b) TP-Link +1.0.1e +1.0.1l +CVE-2013-0166 +CVE-2014-3508 +CVE-2014-8275 +CVE-2015-0286 +CVE-2015-0287 +CVE-2015-0288 +CVE-2015-0289 +CVE-2015-0292 +CVE-2015-1788 +CVE-2015-1790 +CVE-2016-0797 +CVE-2016-2105 +CVE-2016-2176 +CVE-2016-2180 +CVE-2016-2182 +CVE-2016-2160 +14 +0 +4 +0 +14 +0 +14 +38 +14 +38 +4 +23 +14 +38 +14 +0 +14 +38 +14 +38 +14 +38 +14 +38 +14 +33 +14 +38 +4 +13 +4 +28 +0 +5 +10 +15 +20 +25 +30 +35 +(c) Hikvision +Fig. 13. The Distribution of CVEs for Different OpenSSL Versions in vendors Netgear, TP-Link, Hikvision from left to right. +vulnerable functions rank second. 𝑉𝑂2 is performed on 47 detection results, of which 26 are confirmed. Since they +contain the majority of functions, three vendors, Netgear, Tp-Link, and Hikvision, account for the vast majority (94.4%) +of vulnerable functions. Specifically, a large proportion of vulnerable functions are found in the OpenSSL software used +by the three vendors. The number of vulnerable software is consistent with this circumstance. The final column shows +the number of firmware containing at least one vulnerable function, together with its proportion of total firmware. +Every Dajiang firmware contains at least one CVE vulnerability because all OpenSSL components used in firmware are +vulnerable. In addition, Hikvision is detected to have a large proportion of vulnerable firmware (58.89%). To inspect the +CVE vulnerable function distribution, we plot the top 10 CVEs and their distributions in five vendors except Cisco in +Figure 12, since Cisco takes additional two CVEs. +• Top 10 CVE Analysis. Figure 12 demonstrates the top 10 CVE distribution in various vendors. The total number +of discovered CVE vulnerabilities decreases from left to right along the x-axis. Except for CVE-2015-0287, all +of the top 10 CVE vulnerabilities are discovered in every Dajiang firmware. This is because Dajiang utilizes +an outdated version of OpenSSL 1.0.1h that contains numerous vulnerable functions [12]. Although Hikvision +firmware has the third largest number of firmware, it has the most vulnerable functions in our experiment +settings. The reason for this is that Hikvision firmware heavily uses OpenSSL-1.0.1e (184) and OpenSSL-1.0.1l +(401) versions, both of which contain a large number of vulnerabilities. Finding: Since they typically adopt the +same vulnerable software version, it is highly plausible that firmware from the same vendor and released at +the same period contains identical vulnerabilities. Security analysts can quickly narrow down the vulnerability +analysis based on the firmware release date. +• CVE and Version Analysis. Figure 13 depicts the distribution of vulnerable OpenSSL versions for various +CVEs from various vendors. where the x-axis represents the version and the y-axis represents the CVE ID +associated with the vulnerability. Each square in each subfigure indicates the number of OpenSSL versions that +are vulnerable and contain the corresponding CVE along the y-axis. The number is greater the lighter the red +colour. The left subfigure demonstrates that OpenSSL 1.0.2h is widely used by Netgear, resulting in a significant +number of CVE-2016-2180 vulnerabilities (92). Additionally, OpenSSL version 1.0.1e exposes the majority of +CVEs listed on the y-axis, which may increase the device’s attack surface. The TP-Link firmware incorporates +Manuscript submitted to ACM + +Asteria-Pro +27 +OpenSSL version 1.0.1e, resulting in brighter hues. Hikvision firmware utilizes versions 1.0.1e and 1.0.1l, which +are vulnerable to a number of CVEs. Comparing vulnerability distribution in OpenSSL version 1.0.1e among +different vendors reveals inconsistencies in the existence of vulnerabilities. For instance, CVE-2016-2106 is +present in OpenSSL 1.0.1e from Hikvision but not from Netgear and TP-Link. Finding: Despite using the same +version of software, various vendor firmware behaves differently in terms of vulnerability since they can tailor +the software to the device’s specific capabilities. +• CVE-2016-2180 Analysis. The CVE-2016-2180, which is a remote Denial-of-Service flaw caused by received +forged time-stamp file, impacting OpenSSL 1.0.1 through 1.0.2h, exists in 207 firmware. NETGEAR is responsible +for 117 of these, as it deploys 92 OpenSSL 1.0.2h out of a total of 548 firmware. NETGEAR incorporated an +extra nine OpenSSL 1.0.2 series software and sixteen OpenSSL 1.0.1 series software. The vulnerable version +1.0.2h was released in May 2016, and by comparing their timestamps, we determined that OpenSSL 1.0.2h was +integrated into firmware between 2016 and 2019. Finding: Even after their vulnerabilities have been discovered, +the vulnerable versions of software continue to be used for firmware development. +Based on the confirmation results, Asteria-Pro manages to detect 1,482 vulnerable functions out of 1617 bug +search results, indicating that Asteria-Pro achieves a high vulnerability detection precision of 91.65% under our +experiment settings. By randomly selecting 1,000 of 5,604 bug search results, we manually validate the existence of +vulnerabilities in software binaries in order to calculate the recall. Among 1000 bug search results, 205 target function +are confirmed to be vulnerable by checking software versions and the vulnerable functions. Targeting 205 vulnerable +functions, Asteria-Pro detects 53 of them, representing a recall rate of 25.85%. +Finding Inlined Vulnerable Code. During the analysis of mismatched cases, in which the target homologous +functions are not in the top ranking position, we observe that the top-ranked functions contain the same vulnerable +code. We use CVE-2017-13001 as an illustration of inlined vulnerable code detection. CVE-2017-13001 is a buffer +over-read vulnerability in the Tcpdump nfs_printfh function prior to version 4.9.2. After a confirmation operation +𝐶𝑂2, Asteria-Pro reports a single function, parsefh as being vulnerable. We manually compare the decompiled +code of the parsefh function to the source code of nfs_printfh in tcpdump version 4.9.1 (i.e., vulnerable version). +Figure 14 demonstrates that the source code of nfs_printfh (on the left) and the partial code of parsefh (on the right) +are consistent. We designate codes with apparently identical semantics with distinct backdrop hues. In other words, +during compilation, function nfs_printfh is inlined into function parsefh. As a result, the function parsefh contains +CVE-2017-13001 vulnerable code, and Asteria-Pro manages to identify the inlined vulnerable code. Asteria-Pro +has detected an additional eight instances of inlined vulnerable code out of 20 functions in vulnerable circumstance +𝑉𝐶2. +The preceding analysis and conclusions are constrained by the dataset we constructed, which offers security analysts +some recommendations for the security analysis of firmware. +Answer to RQ4. We employ 90 CVE vulnerabilities to search for bugs in 3,483,136 real firmware functions. +Asteria-Pro detects 1,482 vulnerable functions with a high level of precision of 91.65%. In addition, the +capability of Asteria-Pro to identify inlined vulnerable code is stated and illustrated in detail. In conclusion, +Asteria-Pro generates bug search results with a high degree of confidence, thereby reducing analysis labor +by a substantial margin. +Manuscript submitted to ACM + +28 +Shouguo Yang, Chaopeng Dong, and Yang Xiao, et al. +1 +{ +2 +... +3 +if (ndo ->ndo_uflag) { +4 +u_int i; +5 +char const *sep = ""; +6 +ND_PRINT((ndo, " fh[")); +7 +for (i=0; i ψn and φq +n → 0 if πq +n < ψn. The asymptotic behavior of φq +n indicates that it is reasonable +to adopt a threshold-based strategy for solution refinement. Meanwhile, considering the device +sparsity in (7), an element selection operation is necessitated to enforce all the elements except +the one with the largest magnitude in each Xn to be zeros. Consequently, the proposed threshold- +based strategy should be able to perform the following two operations. +Element Selection Operation: To surely guarantee the sparsity constraint in (11), we choose +the largest row in each Xn = [x1 +n, x2 +n, · · · , xQ +n ] and define the index of the largest element as +i∗ +n = arg max +i +xi +n +Hxi +n, ∀n ∈ N. +(35) +Threshold-based Decisive Operation: After obtaining i∗ +n, the binary variable vector αn = +{α1 +n, · · · , αQ +n } can be given as +αn = +� +� +� +ei∗n, +if κi∗ +n +n > 0; +0, +otherwise, +(36) + +12 +where ei∗n is a one-hot vector of length Q with only the i∗ +nth element equal 1 and the others +equal 0, and the corresponding threshold is computed using (31) and (32) as +κi∗ +n +n = +zi∗ +n +n +Hzi∗ +n +n βn +τ 2 +t (βn + τ 2 +t )M − log +� +1 + βn +τ 2 +t +� +. +(37) +3) Limitation: Although the traditional AMP-based algorithm can successfully recover aq +n +from Y , it has some inherent limitations: (i) The traditional AMP algorithm implicitly assumes +Xn has a prior distribution with i.i.d. entries, which neglects the dependencies among the rows +of Xn imposed by the device-level sparsity; (ii) The calculation of the denoiser ηt,n(·) and the +threshold κn requires the exact value of βn, which is costly to obtain in a large-scale mMTC +system with massive devices. +AMP +layer 1 +AMP +layer 2 +AMP +layer t +AMP +layer T +. . . +. . . +. . . +. . . +. . . +. . . +AMP Layers +Refinement Module +… +… +Fig. 2. Network architecture of the proposed DL-mAMPnet. +IV. DEEP LEARNING MODIFIED AMP NETWORK +To address the aforementioned limitations, we propose a deep learning modified AMP network +(DL-mAMPnet). The DL-mAMPnet is constructed by unfolding the AMP algorithm into a +feedforward DNN, which inherits the mathematical model and structure of the AMP algorithm, +thereby avoiding the requirements for accurate modeling. On this basis, we introduce a few +trainable parameters into the DL-mAMPnet to learn the active probability and the large-scale +fading. By making the active probability trainable, we compensate for the inaccuracy caused +by the i.i.d. assumption in the traditional AMP algorithm. By making the large-scale fading +coefficient trainable, we bypass the statistical measurements for the large-scale fadings of mas- +sive devices. According to the threshold-based strategy in Section III-C, we further design a +refinement module to guarantee the device-level sparsity and obtain the desired aq +n. + +13 +As depicted in Fig. 2, the proposed DL-mAMPnet consists of T uniform AMP layers and +one refinement module. For the sake of clarity, each part of the DL-mAMPnet is elaborated +respectively in the following subsection. +A. Input and Output +To facilitate the learning process of DL-mAMPnet, the complex matrices need to be converted +into the real domain and then vectorized. To do this, we first express (8) as +� +� ℜ(Y ) +ℑ(Y ) +� +� = +� +� ℜ(S) +−ℑ(S) +ℑ(S) +ℜ(S) +� +� +� +� ℜ(X) +ℑ(X) +� +� + +� +� ℜ(N) +ℑ(N) +� +� , +(38) +where ℜ(·) and ℑ(·) denote the real and imaginary parts, respectively. The real and imaginary +parts are then concatenated together and vectorized as +˜Y = vec([ℜ(Y )T, ℑ(Y )T]T) ∈ R2LM×1, +(39) +˜S = +� +[ℜ(S), −ℑ(S)]T, [ℑ(S), ℜ(S)]T�T ⊗ IM ∈ R2LM×2NQM, +(40) +˜ +X = vec([ℜ(X)T, ℑ(X)T]T) ∈ R2NQM×1, +(41) +˜ +N = vec([ℜ(N)T, ℑ(N)T]T) ∈ R2LM×1, +(42) +where vec(·) is the vectorize operation that flattens a matrix into a vector in the order of columns, +and ⊗ is the Kronecker product operator. Consequently, (8) can be rewritten as +˜Y = ˜S ˜ +X + ˜ +N. +(43) +According to the recursive formula in (14)-(15), the input to the DL-mAMPnet is chosen to +be the the received signal, the estimated signal, and the residual, which are initialized as ˜ +X0 = 0 +and ˜R0 = ˜Y . Meanwhile, unlike the existing AMP-inspired network that uses ˜ +X [30], we adopt +α = [α1 +1, · · · , αQ +1 , α1 +2, · · · , αQ +N]T ∈ {0, 1}NQ×1 as the output of DL-mAMPnet, such that αq +n can +be directly obtained once DL-mAMPnet is well-trained. + +14 +× +× +- +× +× +Fig. 3. Detailed structure of the tth AMP layer. +B. AMP Layer +Since each layer has the same structure, we focus on the tth AMP layer of the DL-mAMPnet, +of which the detailed structure is illustrated in Fig. 3. Define the input as ˜ +Xt−1, ˜Rt−1 and the +output as ˜ +Xt, ˜Rt, the tth AMP layer proceeds as follows +˜ +Xt = ηt( ˜ +Xt−1 + Bt ˜Rt−1; Θt), +(44) +˜Rt = ˜Y − At ˜ +Xt + +˜Rt−1 +LM +2NQM +� +j=1 +[ηt( ˜ +Xt−1 + Bt ˜Rt−1; Θt)] +′ +j, +(45) +where At and Bt are trainable matrices that acts as the matched filter and Θt = {θt,1, θt,2} is +the trainable parameter set of ηt(·). +It should be mentioned that the denoiser in (28)-(32) cannot be applied in the AMP layer, as +the complex-to-real transformation and vectorization in (39)-(43) have changed the dimension +and distribution of the corresponding matrices. Following the same derivation in Appendix A +but considering ˜ +X as a real-valued Bernoulli Gaussian variable and changing the dimension, +ηt(·) in (28) can be expressed as +[ηt( ˜Z)]j = +β ˜Zj +(β + τ 2 +t ) +� +1 + Q−ϵ +ϵ +exp(log(1 + β +τ 2 +t )1/2 − +˜ +Z2 +j β +2(β+τ 2 +t )τ 2 +t ) +�, += +˜Zj +(1 + τ 2 +t +β ) +� +1 + +� +1 + β +τ 2 +t exp(log( Q−ϵ +ϵ ) − +˜ +Z2 +j +2(τ 2 +t +τ 4 +t /β)) +�, +(46) +where ˜Zj is the jth element of ˜Z. +As discussed in Section II-D, ηt(·) exploits an i.i.d. assumption that fails to effectively explore +the correlated sparsity pattern. To tackle this issue, we replace log(Q−ϵ +ϵ ) with a trainable parameter +θt,1 = [θt,1,1, · · · , θt,1,2NQM]T ∈ R2NQM×1, such that the correlation among entries of ˜ +X can be + +15 +learned and approximated. Meanwhile, to circumvent the need for the prior information of the +large-scale fading, we introduce a trainable parameter θt,2 = [θt,2,1, · · · , θt,2,2NQM]T ∈ R2NQM×1 +and substitute it for β in (46). The trainable ηt(·) can then be defined as +[ηt( ˜Z)]j = +˜Zj +(1 + +τ 2 +t +θt,2,j ) +� +1 + +� +1 + θt,2,j +τ 2 +t +exp(θt,1,j − +˜ +Z2 +j +2(τ 2 +t +τ 4 +t /θt,2,j)) +�. +(47) +The derivative of ηt(·) is thus be given by +[ηt( ˜Z)] +′ +j = [ηt( ˜Z)]j +∂ ˜Zj += +1 + +� +1 + θt,2,j +τ 2 +t +exp(θt,1,j − +˜ +Z2 +j +2(τ 2 +t +τ 4 +t /θt,2,j))(1 + +˜ +Z2 +j +(τ 2 +t +τ 4 +t /θt,2,j)) +(1 + +τ 2 +t +θt,2,j ) +� +1 + +� +1 + θt,2,j +τ 2 +t +exp(θt,1,j − +˜ +Z2 +j +2(τ 2 +t +τ 4 +t /θt,2,j)) +�2 . +(48) +Note that to evade the computation of the expectation involved in τ 2, this paper adopts an +empirical result where τ 2 is estimated by the standard deviation of the corrupted noise in ˜Z, +i.e., τ 2 +t = || ˜Rt||2/ +√ +2LM [30]. +Remark 1: It is worth noting that the denoiser derived in (28) operates in a section-wise +manner, i.e., acts on Q rows of each Xn, while the ηt(·) in the AMP layer operates row-by-row +on X. Although the section-wise manner may exploit the correlations better than the row-wise +manner, it is quite challenging to be implemented in DNNs. This is because to realize such +section-wise manner, we have to either construct N sublayers or impose N iterations in each +AMP layer. The former will heavily expand the network size and trainable parameters, reducing +the scalability and stunting the training process of the DL-mAMPnet. The latter will greatly +increase the computational complexity of the DL-mAMPnet and negate the “deep unfolding” +advantage. It should also be noted that although the AMP layer can explore the correlated +sparsity pattern with the help of trainable parameters, the device-level sparsity constraint in (7) +is not surely guaranteed. Motivated by this consideration, we propose a felicitous method in +the refinement module that utilizes the Maxpool-MaxUnpool operation to ensure device-level +sparsity, as detailed in the subsection below. +C. Refinement Module +The refinement module should be capable of ensuring the device-level sparsity while extracting +aq +n from +˜ +XT without explicit channel state information (CSI). To fulfil these functionalities, +two components are integrated in the refinement module, namely the soft-thresholding denois- +ing component and the hard-thresholding decision component. The soft-thresholding denoising +component is intended to further denoise ˜ +XT by exploiting the hierarchical sparse structure. The + +16 +× +- +- +2NQ× M +2NQM × 1 +2NQ × 1 +2NQ × 1 +2NQ × 1 +2NQ × 1 +2NQ × 1 +2NQ × 1 +2NQ × 1 +2N × 1 +2NQ × 1 +2NQM × 1 +1 × 1 +(2NQM+1) × 1 +2NQ × 1 +NQ × 1 +Reshape +Absolute +Conv +FC+ReLU +FC+ReLU +FC+ReLU +Sigmoid +Average +ReLU +MaxPool +Concatenate +MaxUnpool +FC+ +Soft-thresholding Denosing +Hard-thresholding Decision +Fig. 4. Detailed architecture of the proposed refinement module. +hard-thresholding decision component is aimed at implementing the threshold-based strategy in +(35)-(37). The detailed structure of the refinement module is presented in Fig. 4 and elaborated +as follows. +Soft-Thresholding Denoising: As shown in Fig. 1, the two-level sparsity exhibits a unique +spatial structure that has not been utilized in the AMP layers. Here, the soft-thresholding +denoising aims to distill +˜ +XT using such spatial feature, enhancing useful information while +removing noise information. To do this, we first de-vectorize ˜ +XT and take the absolute value as +X = |Vec−1( ˜ +XT)| = [|ℜ(X)T|, |ℑ(X)T|]T ∈ R+2NQ×M. +(49) +Then, a convolutional layer with 1 × M kernel size is applied to X to combine the information +from all M antennas and extract a coarse estimation of aq +n. This arrangement is motivated by the +fact that all M elements in each row of X share the same aq +n, as observed from (6) and Fig. 1. +The coarse estimation can be expressed as fθc(X), where fθc(·) is the function expression of the +convolutional layer with parameter θc. After that, an average pooling with 1 × M kernel size is +applied to X to get a 1-D average vector over M antennas. The 1-D vector ι = +1 +M +�M +m=1 X:,m +is forwarded into a two-layer fully-connected (FC) network to obtain a scaling parameter, such +that the inner features of the average value among the 2NQ rows of X can be learned. The +scaling parameter is then scaled to the range of (0, 1) using a sigmoid function, which can be +written as follows +ϑ = +1 +1 + e +−fθF C1 (ι), +(50) + +17 +where ϑ is the scaling vector and fθF C1(·) is the function expression of the two-layer FC network +with parameter θFC1. Next, ϑ is multiplied by ι to get the threshold as +κST = ϑ ⊙ ι, +(51) +where ⊙ is the Hadamard product operator. This operation is inspired by the fact that the +threshold for soft thresholding must be positive and not too large [31]. If the threshold is larger +than the largest value of fθc(X), then the output of soft thresholding will all be zeros, and thus +the useful information will be removed. Finally, the obtained threshold κST is subtracted by +fθc(X) and fed into a ReLU activation function as +o = max(0, fθc(X) − κST), +(52) +where o denotes the output of the soft-thresholding denoising component. We can observe from +(52) that by keeping κST in a reasonable range, the useful information can be preserved while +the noise information is eliminated. It is worth noting that, rather than being manually set +by experts, such a threshold can be learned automatically in the proposed soft-thresholding +denoising component, removing the need for the expertise of signal processing and the statistical +characteristic of X. +3 +7 +1 +5 +9 +8 +5 +9 +8 +0 +0 +0 +5 +9 +8 +Pooling Indices +MaxPool +MaxUnpool +Filter +Fig. 5. Illustration of the MaxPool-MaxUnpool process. +Hard-thresholding Decision: It is challenging to directly implement the threshold-based +strategy in DNNs, as (35) is non-differentiable and will stunt the backpropagation process. +To tackle this issue, the hard-thresholding decision component elegantly uses the Maxpool +and MaxUnpool procedures to ensure the device-level sparsity. Maxpool is a down-sampling +technique that uses a max filter to non-overlapping subregions of the initial input [32]. For each +region represented by the filter, we will take the max of that region and create a new output + +18 +matrix where each element is the max of a region in the original input. Maxunpool, in contrast, +expands the output of the maxpool operation to its original size by upsampling and padding +with zeros. Except for the maximum position, all the rest elements in the unpooled matrix are +supplemented with 0. +For an intuitive explanation, we illustrate the process of Maxpool and MaxUnpool in Fig. 5. +It can be observed from Fig. 5 that in each filter, except for the largest value that remains +unchanged, all the rest elements become 0. Such manipulation perfectly executes the element +selection operation in (18). By setting the filter size as Q × 1, we enforce that at most one +non-zero row exists in the Q rows of Xn, and therefore the device-level sparsity constraint in +(11) can be guaranteed. It should also be mentioned that the pooling procedure is only a module +that alters the dimension size during the deep learning process, which has no parameters and +thus has no impact on network training. +After guaranteeing the device-level sparsity, the onus shifts to performing the threshold-based +decisive operation in (36), i.e., determining the binary sequence α by comparing the threshold +κi∗ +n +n with the matrix obtained from the maxpool-maxunpool procedure Mp(Mup(o)). However, +some issues exist when determining α. The first issue is that the threshold in (37) may not be +precise sufficiently because it is derived under an mismatched i.i.d. assumption. To tackle this +issue, we look afresh at (37) and find that the threshold is a function of β and τ. Since β has +been represented by θ2 in (47), we concatenate θT,2 and τ 2 +T outputted from the last AMP layer +and feed it into an FC layer with ReLU activation function to learn the accurate threshold, which +is denoted by +κHT = max(0, fθF C2(θT,2, τ 2 +T)), +(53) +where fθF C2 is the function expression of the FC network with parameter θFC2. +Then, the learned threshold κHT is subtracted by Mp(Mup(o)) and forwarded into an FC +layer with parameter θFC3 to fulfil the threshold-based decisive operation. The FC layer here +has two functionalities: compressing the dimension from 2NQ × 1 to NQ × 1 and converting +the κHT-Mp(Mup(o)) difference into a binary sequence. Mathematically, the optimal function +for threshold-based binary decision is the signum function denoted as +sng(x) = +� +� +� +1, +x > 0; +0, +x ≤ 0. +(54) + +19 +However, since sng(x) is non-differentiable, it cannot be used in DNN, necessitating the devel- +opment of a substitute function. +-10 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +10 +Input +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Output +Region with positive output +and negative input +Sgn (Optimal) +Sigmoid +Proposed (m=1) +Proposed (m=5) +Proposed (m=10) +Fig. 6. The curves of the optimal signum, sigmoid, and hard-thresholding decision functions. +When it comes to DL-based binary decisions, the sigmoid function is a popular choice and +has been widely used in the literature [33], as it can map the input to the interval within [0, 1]. +The sigmoid function, nevertheless, is still inapplicable to the hard-thresholding decision module. +The reasons are as follows: (i) The sigmoid function returns a continuous value between 0 and +1, implying that a threshold is further required to distinguish the outputted value as 0 or 1. +However, it is usually non-trivial to design an appropriate threshold; (ii) According to (36), the +output of the threshold-based decision should be strictly 0 with negative input. However, as +shown in Fig. 6, there is a region where the output is still positive with negative input in the +sigmoid function, which may introduce additional errors. To solve the above issues, we devise a +novel hard-thresholding decision function, whose core idea is to cascade the ReLU function with +tahn function and introduce a multiplier ϱ to approximate the cascaded function as a signum +function. The proposed hard-thresholding decision function is given by +fϱ(x) = max(0, eϱx − e−ϱx +eϱx + e−ϱx). +(55) +By cascading the ReLU function with tahn function, we not only ensure that the output of +the threshold-based decision is strictly 0 with negative input, but also guarantee the output +with positive input approximates to 1 with the increment of ϱ. The optimal signum, sigmoid, +and hard-thresholding decision functions are plotted in Fig. 6. The figure shows that with the +increase of ϱ, fϱ(·) gradually approximates to sng(x), validating the rationality of the proposed +hard-thresholding decision function. + +20 +Remark 2: Although we restrict the application of the hard-thresholding decision component +to the non-coherent transmission in mMTC, the proposed component can be used in any other +scenarios where the signal has a special sparsity structure, such as the spatial modulation system. +Meanwhile, the devised hard-thresholding decision function can also be used in any bit-level +detector. That is, the hard-thresholding decision component is a plug-and-play module with a +wide range of applications. +V. THE IMPLEMENTATION OF DL-MAMPNET +A. Parameter Initialization +In deep learning, parameter initialization plays a critical role in speeding up convergence and +achieving lower error rates. Choosing proper initialization values is especially important for the +proposed DL-mAMPnet, as the DL-mAMPnet is built on the AMP algorithm and thus should +preserve some essential features to ensure performance and interpretability. There are mainly +three items needed to be considered for parametrization: the trainable matrices At and Bt, the +denoiser parameter set Θt, and the refinement module parameters θRM = {θFC1, θFC2, θFC3, θC}. +1) Initializing At and Bt: It can be observed from (44)-(45) that the DL-mAMPnet imple- +ments a generalization of the AMP algorithm in (14)-(15), wherein the matched filters (S, SH +n ) +manifest as (At, Bt) at iteration t. However, such generalization does not enforce Bt = AH +t and +thus may not preserve the independent-Gaussian nature of the denoiser input (19). According to +the analysis in [30], the desired nature maintains when At = υtS with υt > 0. Therefore, At is +parameterized as υtS and (44)-(45) can be rewritten as +˜ +Xt = υtηt( ˜ +Xt−1 + Bt ˜Rt−1; Θt), +(56) +˜Rt = ˜Y − S ˜ +Xt + υt ˜Rt−1 +LM +2NQM +� +j=1 +[ηt( ˜ +Xt−1 + Bt ˜Rt−1; Θt)] +′ +j, +(57) +the derivation of which can be found in [30] and is omitted here for brevity. In this paper, we +initialize Bt = ˜ST and υt = 1, since such initialization can greatly expedite the convergence of +the training process [30]. +2) Initializing Θt: For θ1, we initialize each element as log(Q−ϵ +ϵ ), i.e., initialize that each +pilot sequence has the same active probability. This is because we have no prior information +about the device activity and the transmitted pilot sequence index. By adopting such a uniform +initialization, the initial θ1 will have the minimum Euclidean distance from the actual value. + +21 +For example, consider a device with a 2-bit message and active indicator {1, 0, 0, 0}. If we start +with a mismatched one-hot vector, then the Euclidean distance will be +√ +2. If we initialize αn +as {1 +4, 1 +4, 1 +4, 1 +4}, then the Euclidean distance will be +� +3 +4. Therefore, the uniform initialization can +accelerate the convergence as a shorter Euclidean distance may lead to faster convergence. +The initial value of θ2 can be computed from the received signal strength. Recall that each +pilot sequence has a unit norm and hn ∼ CN(0, βnIM), each element of the initial θ2 is roughly +given by || ˜Y ||2 +2/ +√ +2K. +3) Initializing θRM: For all parameters in the refinement module, we adopt the He initial- +ization [34] as it has been mathematically proved to be the best weight initialization strategy for +the ReLU activation function [35]. +B. Parameter Training +1) Training Algorithm: Aside from the network structure and parameter initialization, the +training algorithm also determines the performance of the DL-mAMPnet. The standard training +strategy is the end-to-end training where all the parameters are optimized simultaneously by +following the back-propagation rule. However, the end-to-end training is not appropriate for the +DL-mAMPnet due to the following reasons: (i) The AMP algorithm aims to provide an estimate +ˆ +X(Y ) based on Y that minimizes the MSE EXY || ˆ +X(Y )−X||2 +2. If the DL-mAMPnet is trained +to learn the direct mapping from Y to α, the MSE optimality of the AMP layers may not be +achieved; (ii) Even if the AMP layers and the refinement module are trained separately, the AMP +layers can still easily converge to a bad local optimal solution due to overfitting [36]. +For these reasons, we propose a layer-wise training strategy, the idea behind which is to +decouple the training of each layer. The details are given in Algorithm 1. There are totally +T + 2 phases in the layer-wise training. In the first phase, we train the learnable parameters of +the first AMP layer. Then in the t phase, we train the first t AMP layers with the parameters +of the first t − 1 AMP layers fixed as the parameters learned by the first t − 1 phases. In the +T + 1 phase, we train the whole network with only the parameters of the refinement module +is learnable, while the parameters of the AMP layers are fixed as the parameters learned by +the first T phases. Finally, in the last phase, all the parameters are initialized as the parameters +learned during the first T + 1 phases and then trained jointly. + +22 +Algorithm 1 Parameter training of the DL-mAMPnet via layer-wise training strategy +Input: Training dataset DAMP, DRM; +Output: Trained parameter {υt, Bt, Θt}T +t=1 and θRM; +Initialize parameters according to Section IV-B; +for t = 1 to T do +Learn {υt, Bt, Θt}t with fixed {υt, Bt, Θt}t−1 +t=1 based on the loss function (58); +end for +Learn θRM with fixed {υt, Bt, Θt}T +t=1 based on the loss function (59); +Re-learn {υt, Bt, Θt}T +t=1 and θRM based on the loss function (59); +return {υt, Bt, Θt}T +t=1 and θRM. +The training dataset DAMP for the first T phases comprises 100, 000 pairs of ˜ +X and ˜Y , and +the corresponding loss function is the MSE loss +Lt( ˜Y ) = || ˜ +Xt( ˜Y ) − ˜ +X||2 +2, t = [1, · · · , T]. +(58) +The training dataset DRM for the last 2 phases has 100, 000 pairs of α and ˜Y , and the loss +function is the binary cross entropy loss +Lt( ˜Y ) = +1 +NQ +NQ +� +i=1 +� +α( ˜Y )i log αi + (1 − α( ˜Y )i) log(1 − αi) +� +, t = [T + 1, T + 2]. +(59) +The DL-mAMPnet is trained epoch by epoch with the training dataset using the Adam optimizer, +while within an epoch, the whole training dataset is shuffled and split into batches with the size +of 500.1 +2) Training Dataset: The training dataset is synthetically generated as follows: (i) Generating +αn: K active devices are randomly selected among N devices. Then, each active device is +randomly assigned with a Q-dimensional one-hot vector, and each inactive device is assigned +with a Q-dimensional zero vector; (ii) Generating Xn: The uplink channel of device n, i.e., hn, is +first generated according to (1). Then Xn is obtained by multiplying hn and αn; (iii) Generating +Y : The pilot sequence Sn is generated by sampling from complex Gaussian distribution with +zero mean and variance. Given Xn and Sn, Y can be directly obtained according to (6). +1It should be mentioned that the number of epochs and the learning rate are different for each phase, which are empirically +determined in Section V. + +23 +VI. SIMULATION RESULTS +In this section, extensive simulations are provided to verify the effectiveness of the proposed +algorithm. The setup is as follows unless otherwise stated. We consider a mMTC system with +N = 100 devices for illustration purpose, although the proposed algorithm can be used for +a much larger-scale system. Each device accesses the BS independently with probability ϵ = +0.1 at each coherence block. The large-scale fading coefficient for device n is βn = 128.1 − +36.7 log10(dn) in dB, where dn is the distance between device n and the BS that follows a uniform +distribution within [0.05, 1] km. The small-scale fading coefficient for each device follows the +i.i.d. multivariate complex Gaussian distribution with zero mean and unit variance. The power +spectral density of the AWGN at the BS is assumed to be −169 dBm/Hz [8] and the bandwidth +of the wireless channel is 1 MHz. +The number of AMP layers in the DL-mAMPnet is set to be T = 4. The training epochs and +learning rate for each training phase are set to be {2, 000, 1, 500, 1, 000, 1, 000, 1, 500, 5, 000} and +{2 × 10−5, 2 × 10−5, 2 × 10−5, 2 × 10−5, 1 × 10−5, 1 × 10−5}.2 We train the DL-mAMPnet with +80, 000 training samples and test with 20, 000 data samples, which are randomly drawn from +DAMP for the first 4 phases and DRM for the last 2 phases. The DL-mAMPnet is trained and +tested by on an x86 PC with one Nvidia GeForce GTX 1080 Ti graphics card, and Pytorch 1.1.0 +is employed as the backend. The traditional AMP-based algorithm with TAMP = 50 iterations +and the covariance-based method with TCov = 50 iterations [16] are employed as the benchmark +and evaluated on the same dataset. In addition, the SER is adopted as the performance metric: +SER = 1 +N +�N +n=1 I( ˆαn ̸= αn), where ˆαn and αn denote the estimated pilot sequence activity for +device n and its ground truth, respectively. +A. Performance of the DL-mAMPnet +Fig. 7 depicts the SER versus L with different values of M. It is observed that both the SER +of the DL-mAMPnet and AMP-based algorithm decrease as L and M increase. Although the +SER of the covariance-based algorithm is lowest when L is small, it becomes saturated when L +exceeds some point, e.g., L = 40 when M = 16. This is mainly due to the suboptimality of the +2All the parameters are empirically determined using the general workflow, where the training starts with relatively small +values and increases the values until the learning performance cannot be further improved. + +24 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Pilot Sequence Length: L +10-4 +10-3 +10-2 +10-1 +100 +SER +AMP, M=8 +AMP, M=16 +AMP, M=32 +Covariance, M=8 +Covariance, M=16 +Covariance, M=32 +DL-mAMPnet, M=8 +DL-mAMPnet, M=16 +DL-mAMPnet, M=32 +Fig. 7. SER performance versus the pilot sequence length L for J = 1 bit. +0 +5 +10 +15 +20 +25 +30 +35 +40 +Number of Receiving Antennas: M +10-4 +10-3 +10-2 +10-1 +100 +SER +AMP, L=50 +AMP, L=60 +AMP, L=70 +Covariance, L=50 +Covariance, L=60 +Covariance, L=70 +DL-mAMPnet,L=50 +DL-mAMPnet,L=60 +DL-mAMPnet,L=70 +Fig. 8. SER performance versus the number of receiving antennas M for J = 1 bit. +fixed threshold.3 Meanwhile, the proposed DL-mAMPnet notably outperforms the AMP-based +algorithm by a large margin. For example, the proposed DL-mAMPnet achieves more than 10 +pilot length gain over the AMP-based algorithm when L is larger than 70, which indicates that +the proposed DL-mAMPnet can reduce the required pilot sequence length, lowering the difficulty +of pilot design and adapting to fast-changing channels. Moreover, although for any M, the SERs +of both the DL-mAMPnet and AMP-based algorithm decrease over L, the reduction is faster +3As observed from (37), the threshold is variable and related to system parameters such as signal power and receiving antenna +numbers, whereas the covariance-based algorithm adopts a fixed threshold. Since there is no concrete method to design such a +fixed threshold, we empirically set the threshold of the covariance-based algorithm to be βn/2 in this paper. + +25 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Pilot Sequence Length: L +10-3 +10-2 +10-1 +100 +SER +AMP, M=16,J=1bit +AMP, M=16,J=2bits +Covariance, M=16,J=1bits +Covariance, M=16,J=2bits +DL-mAMPnet, M=16,J=1bit +DL-mAMPnet,M=16,J=2bits +Fig. 9. SER performance versus the pilot sequence length L with different lengths of transmitted messages J. +when M is 32 as compared to that when M is 8, which shows that increasing the number of +receiving antennas can further reduce the required pilot sequence length. +Fig. 8 shows the SER versus M for various values of L. We observe that for the DL-mAMPnet +and AMP-based algorithm, the SER drops effectively as M increases, whereas for the covariance- +based algorithm, there are error floors in the SER. Moreover, the DL-mAMPnet needs fewer +receiving antennas to achieve the same performance as the AMP-based algorithm, implying +that the proposed DL-mAMPnet can reduce demand for receiving antennas, resulting in lower +deployment cost and energy consumption. +Fig. 9 plots the SER versus L, with 2 different lengths of transmitted messages, i.e., J = 1 bit +and J = 2 bits. The number of receiving antennas is M = 16. It can be seen that the SERs of all +three algorithms increase as the length of transmitted messages increases, which implies that the +performance of both algorithms deteriorates when more messages are transmitted. An important +point is that as the message length increases, the performance gap between the proposed DL- +mAMPnet and the other two algorithms increases, indicating the potential of the DL-mAMPnet +to handle long packet size. +B. Visualization of the DL-mAMPnet +To offer more insights of the proposed DL-mAMPnet, we present the visualization of the +outputs of each component of a well-trained DL-mAMPnet. For clarity, we only present the +case where N = 10 devices transmit 1-bit message with ϵ = 0.1 active probability, L = 10 pilot + +26 +0123 +0 +4 +8 +12 +16 +20 +24 +28 +32 +36 +(a) +0123 +0 +4 +8 +12 +16 +20 +24 +28 +32 +36 +(b) +0 +0 +4 +8 +12 +16 +20 +24 +28 +32 +36 +(c) +0 +0 +4 +8 +12 +16 +(d) +0 +0 +4 +8 +12 +16 +(e) +Fig. 10. +A visualization of a well-trained DL-mAMPnet. (a) ˜ +XT , the output of AMP layers; (b) The ground truth ˜ +X; (c) +Mp(Mup(o)), the output of the maxpool- maxunpool procedure; (d) ˆα, the output of the refinement module; (e) The ground +truth α. +sequence length, and M = 2 receiving antennas. For visualization, we transform the outputs of +each component to the reverse grayscale images. Specifically, the elements of each output matrix +are normalized to an interval within [0, 1], where 0 and 1 are represented by white color and +black color, respectively. It should be mentioned that we take the absolute value of ˜ +X and ˜ +XT +to show the signal strength difference more intuitively. +The output of the AMP layers and its ground truth are shown in Fig. 10(a) and Fig. 10(b), +respectively. It can be seen that the non-zero rows of ˜ +X are correctly recovered, paving the way +for the subsequent refinement progress. Then, the output of the maxpool-maxunpool procedure +is visualized in Fig. 10(c), where the largest of the two adjacent rows is retained and the other +becomes 0, demonstrating the validity of the maxpool-maxunpool procedure in ensuring the +device-level sparsity. Fig. 10(d) and Fig. 10(e) are the visualizations of ˆα and α, where we +find that the pilot sequence activity is perfectly estimated by the well-trained DL-mAMPnet. +Moreover, it is observed from Fig. 10(c) and Fig. 10(e) that the pilot sequence activity is correctly +reserved in Fig. 10(c) (the 1st, 4th, 21st, and 24th rows), which indicates the effectiveness of +the proposed soft-thresholding denoising component. +C. Computational Complexity Analysis +Finally, we analyze the computational complexities of the traditional AMP-based algorithm +and DL-mAMPnet. + +27 +For the traditional AMP-based algorithm, the computational complexity mainly comes from +the matrix multiplication in (14)-(15) [8]. Since SH +n +∈ CQ×L, Rt ∈ CL×M, S ∈ CL×NQ, +and Xt+1 ∈ CNQ×M, the computational complexity for N devices and TAMP iterations is +O(4TAMP(NQLM + NQLM)) = O(8TAMPNQLM), where the proportional constant “4” +appears because a complex multiplication requires 4 real multiplications, the former “NQLM” +comes from the multiplication between SH +n and Rt for N devices and the latter “NQLM” +comes from the multiplication between S and Xt+1. After the iterative process, the AMP-based +algorithm requires the element selection operation (i.e., (35)) whose computational complexity +is O(4NQM), and the threshold calculation (i.e., (37)) whose computational complexity is +O(4NQM). Taking all the operations into account, the computational complexity of the AMP- +based algorithm is given by O(8TAMPNQLM). +For the proposed DL-mAMPnet, we focus on the computational complexity of online imple- +mentation. The computational complexity of the AMP layers comes from the matrix multipli- +cation Bt ˜Rt−1 and At ˜ +Xt, which is O(8TDLNQLM 2) with TDL denoting the number of AMP +layers. For the refinement module, the computational complexity is mainly resulted from the FC +and convolutional layers. For a FC layer with Nl−1 input and N1 output, its computational +complexity is given by O(Nl−1N1). For a convolutional layer with a H × W input and a +Hf × Wf filter, its computational complexity can be expressed as O(HWHfWf). Therefore, +the total computational complexity of the refinement module is O(4N 2Q2M). Consequently, the +computational complexity of DL-mAMPnet is O(8TDLNQLM 2 + 4N 2Q2M). +From the above discussions, it seems that the proposed DL-mAMPnet can achieve better +performance at the expense of a higher computational complexity compared to the AMP-based +algorithm. However, as observed in Fig. 7-Fig. 9, the DL-mAMPnet with TDL = 4 AMP layers +outperforms the AMP-based algorithm with TAMP = 50 iterations, indicating that the proposed +DL-mAMPnet may need less computational complexity to achieve the same SER performance +with the AMP-based algorithm. +VII. CONCLUSION +This paper has proposed a novel DL-based algorithm, termed DL-mAMPnet, for the joint +device activity and data detection in mMTC with a single-phase non-coherent scheme. Trainable +parameters have been added in the DL-mAMPnet to compensate for the inaccuracy caused by +the i.i.d. assumption in the traditional AMP algorithm. A refinement module has been further + +28 +designed to enhance the SER performance and guarantee the device-level sparsity by exploiting +the correlated sparsity pattern. The proposed algorithm can be applied to scenarios where massive +users intermittently transmit small packets, e.g., smart home and industrial control. For the future +work, we will investigate the pilot sequence design scheme to maintain orthogonality and mitigate +the inter-device interference. +APPENDIX A +DERIVATION OF MMSE DENOISER (21) +To enable the derivation of the conditional probability PXn|Zn, we assume xq +n is independent +with each other, and thus we have +Pxq +n = +� +1 − ϵ +Q +� +δ + ϵ +Q +exp(−xq +n +H(βnIM)−1xq +n) +πM|βnIM| +. +(60) +According to (20), the likelihood of observing zq +n given xq +n is +Pzq +n|xq +n = exp(−(zq +n − xq +n)HΣ−1(zq +n − xq +n)) +πM|Σ| +. +(61) +Denoting k as the proportional constant, Pxq +n|zq +n can be computed using the Bayes’ formula +as follows +Pxq +n|zq +n = kPzq +n|xq +nPxq +n += k +� +(1 − ϵ +Q)δ + ϵ +Q +exp(−xq +n +H(βnIM)−1xq +n) +πM|βnIM| +� �exp(−(zq +n − xq +n)HΣ−1(zq +n − xq +n)) +πM|Σ| +� += k +� +(1 − ϵ +Q)exp(−zq +n +HΣ−1zq +n) +πM|Σ| +δ + ϵ +Q +exp(−xq +n +H(βnIM)−1xq +n − (zq +n − xq +n)HΣ−1(zq +n − xq +n)) +π2M|βnIM||Σ| +� +. +(62) +Note that +xq +n +H(βnIM)−1xq +n + (zq +n − xq +n)HΣ−1(zq +n − xq +n) = (xq +n − ζ)HΞ−1(xq +n − ζ) + zq +n +H∆−1zq +n, +where Ξ = ( 1 +βnIM + Σ−1), ζ = ΞΣ−1zq +n, and ∆ = βnIM + Σ, (62) can be rewritten as +Pxq +n|zq +n += k +� +(1 − ϵ +Q)exp(−zq +n +HΣ−1zq +n) +πM|Σ| +δ + ϵ +Q +exp +� +−(xq +n − ζ)HΞ−1(xq +n − ζ) − zq +n +H∆−1zq +n +� +π2M|βnIM||Σ| +� +. +(63) + +29 +Since +� +Pxq +n|zq +n dxq +n = 1, k can be obtained by integrating (63) out. Accordingly, we have +k = +� +(1 − ϵ +Q)exp(−zq +n +HΣ−1zq +n) +πM|Σ| ++ ϵ +Q +exp +� +−zq +n +H∆−1zq +n +� +|Ξ| +πM|βnIM||Σ| +�−1 +(a) += +� +(1 − ϵ +Q)exp(−zq +n +HΣ−1zq +n) +πM|Σ| ++ ϵ +Q +exp +� +−zq +n +H∆−1zq +n +� +πM|∆| +�−1 +, +(64) +where (a) holds because | 1 +βnIM + Σ−1| = |βnIM||Σ|/|βnIM + Σ|. +Substituting (64) into (62), Pxq +n|zq +n can be determined as +Pxq +n|zq +n = e−(xq +n−ζ)HΞ−1(xq +n−ζ)ϵ|Σ| + (Q − ϵ)e−zq +n +H(Σ−1−∆−1)zq +nπM|Ξ||∆|δ +ϵπM|Ξ||Σ| + (Q − ϵ)e−zq +n +H(Σ−1−∆−1)zq +nπM|Ξ||∆| +. +(65) +Hence, the conditional expectation E{xq +n|zq +n} is given by +E{xq +n|zq +n} = +� +xq +nPxq +n|zq +n dxq +n = +ζϵ|Σ| +ϵ|Σ| + (Q − ϵ)e−zq +n +H(Σ−1−∆−1)zq +n|∆| += +βn(βnIM + Σ)−1zq +n +1 + Q−ϵ +ϵ |IM + βnΣ−1|e−zq +n +H(Σ−1−(Σ+βnIM)−1)zq +n . +(66) +The MMSE-optimal denoiser in (21)-(25) can be straightforwardly obtained from (66) through +simple mathematical transformation. +REFERENCES +[1] N. H. 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Schmidhuber, “Multi-column deep neural networks for image classification,” in Proc. IEEE +CVPR, 2012, pp. 3642–3649. +[33] X. Gao et al., “ComNet: Combination of deep learning and expert knowledge in OFDM receivers,” IEEE Commun. Lett., +vol. 22, no. 12, pp. 2627–2630, Dec. 2018. +[34] K. He, X. Zhang, S. Ren, and J. Sun, “Delving deep into rectifiers: Surpassing human-level performance on ImageNet +classification,” in Proc. IEEE ICCV, 2015, pp. 1026–1034. +[35] S. K. Kumar, “On weight initialization in deep neural networks,” arXiv preprint, arXiv:1704.08863, 2017. +[36] C. Metzler, A. Mousavi, and R. Baraniuk,“Learned D-AMP: Principled neural network based compressive image recovery,” +in Proc. NIPS, 2017, pp. 1772–1783. + diff --git a/3dAyT4oBgHgl3EQfo_hK/content/tmp_files/load_file.txt b/3dAyT4oBgHgl3EQfo_hK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d5aea15ae000d21dd7017556e2220847f28964f9 --- /dev/null +++ b/3dAyT4oBgHgl3EQfo_hK/content/tmp_files/load_file.txt @@ -0,0 +1,914 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf,len=913 +page_content='1 Model-Driven Deep Learning for Non-Coherent Massive Machine-Type Communications Zhe Ma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Wen Wu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Senior Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Feifei Gao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Fellow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' and Xuemin (Sherman) Shen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Fellow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' IEEE Abstract In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' we investigate the joint device activity and data detection in massive machine-type communications (mMTC) with a one-phase non-coherent scheme,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' where data bits are embedded in the pilot sequences and the base station simultaneously detects active devices and their embedded data bits without explicit channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Due to the correlated sparsity pattern introduced by the non- coherent transmission scheme, the traditional approximate message passing (AMP) algorithm cannot achieve satisfactory performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Therefore, we propose a deep learning (DL) modified AMP network (DL-mAMPnet) that enhances the detection performance by effectively exploiting the pilot activity correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The DL-mAMPnet is constructed by unfolding the AMP algorithm into a feedforward neural network, which combines the principled mathematical model of the AMP algorithm with the powerful learning capability, thereby benefiting from the advantages of both techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Trainable parameters are introduced in the DL-mAMPnet to approximate the correlated sparsity pattern and the large-scale fading coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Moreover, a refinement module is designed to further advance the performance by utilizing the spatial feature caused by the correlated sparsity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Simulation results demonstrate that the proposed DL-mAMPnet can significantly outperform traditional algorithms in terms of the symbol error rate performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Index Terms Massive machine-type communication (mMTC), non-coherent transmission, grant-free random ac- cess, deep learning, model-driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Ma and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Gao are with the Institute for Artificial Intelligence Tsinghua University, State Key Lab of Intelligent Technologies and Systems, Beijing National Research Center for Information Science and Technology, Department of Automation, Tsinghua University, Beijing 100084, China (e-mail: maz16@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' feifeigao@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Wu is with the Frontier Research Center, Peng Cheng Laboratory, Shenzhen, Guangdong 518055, China (email: wuw02@pcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Shen is with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: sshen@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='ca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='00516v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='IT] 2 Jan 2023 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' INTRODUCTION To embrace the forthcoming era of Internet of Things (IoT), the 3rd Generation Partnership Project (3GPP) has specified massive machine-type communications (mMTC) as one of the three main service classes for fifth-generation (5G) network and beyond [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In a typical mMTC scenario, a massive number of IoT devices are required to establish uplink-dominated commu- nication with a single base station (BS) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The uplink transmission is usually sporadic and has a short packet size, so only a small and random subset of devices are active for a short while [3]- [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' As a result, conventional grant-based random access protocols are inappropriate for the mMTC scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' To better support mMTC services, one potential solution is to develop novel multiple-access schemes that can accomplish user activity and data detection in a timely and accurate manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Grant-free (GF) random access is a promising solution for mMTC and IoT, as it eliminates the signaling overhead required for the coordination between the BS and massive devices [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In the GF-random access, the user activity and data detection are usually conducted through a two-phase coherent scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Specifically, each activated device directly transmits a unique pilot sequence followed by data packets without a prior scheduling assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' After receiving the superimposed signal from these devices, the BS first detects the active devices and estimates the channel, based on which the corresponding transmitted data bits are then decoded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' However, due to the massive number of devices, it is impossible to assign orthogonal pilot sequences to each device, which inevitably leads to collisions among devices and results in performance degradation [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Thanks to the sporadic mMTC traffic pattern, the device activity detection and channel estimation can be formulated as a compressed sensing (CS) problem [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Consequently, various CS techniques have been considered for device detection in mMTC, and they have been shown to outperform traditional methods by mitigating pilot contamination [8]- [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Nevertheless, the two-phase coherent scheme incurs non-negligible overhead for channel training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Thus it may not be suitable for mMTC where devices usually transmit small packets intermittently, prompting researchers to consider the non-coherent schemes [11]- [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Several existing works have attempted to investigate the one-phase non-coherent scheme [13]- [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In contrast to the coherent scheme, explicit channel estimation is not required in the non-coherent scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The intuition behind the one-phase non-coherent scheme is to allocate multiple distinct pilot sequences to each device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' When transmitting, each device selects only 3 one pilot sequence based on its data, and the BS detects the user activity and data jointly by determining which pilot sequence is received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The paper [13] proposes a novel method for embedding 1 bit in pilot sequences, which outperforms the two-phase coherent scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The work [14] considers the case when multiple bits are embedded and conducts joint user activity and data detection using the approximate message passing (AMP) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In [15], a modified- AMP algorithm is proposed, where the soft-thresholding function is utilized to decide on one of the possible pilot sequences while suppressing the other ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In [16], a covariance-based detection scheme is developed to acquire the indices of the transmitted pilot sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' However, all the aforementioned works assume that the activity of each pilot sequence is independently and identically distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Although the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' assumption produces an analytically tractable solution, it neglects the correlation among the pilot sequence activity in each user and thus may not be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In this work, we investigate the possibility of applying the deep learning method to explore the correlation structure of the sparsity pattern and improve the joint user activity and data detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Thanks to the strong capability of solving intricate and intractable problems, machine learning has become a favorable research topic for future wireless communications [17]- [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In particu- lar, as a major branch in machine learning, deep learning has been extensively investigated for sig- nal detection [20], channel estimation [21], and constellation design [22] to improve performance while reducing computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Among vast techniques that employ deep learning in wireless communication, the “deep unfolding” method that unfolds iterative algorithms into deep neural networks (DNN) is especially attractive [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' By incorporating communication expert knowledge into DNN, “deep unfolding” inherits the mathematical models of classic algorithms and enables the interpretation of network topology design [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Meanwhile, by exploiting the powerful learning capability of DL, “deep unfolding” compensates the imperfections resulting from the inaccuracy of the model and predetermined parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Motivated by existing works, we propose a model-driven DL algorithm, namely DL-modified AMP network (DL-mAMPnet), for the joint device activity and data detection in mMTC with single-phase non-coherent scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' DL-mAMPnet is constructed by unfolding the AMP algorithm while adding trainable parameters and a refinement module to explore the correlated sparsity pattern of the pilot sequence activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Simulation results validate the superior symbol error rate (SER) performance of the proposed DL-mAMPnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The main contributions can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 4 We formulate the joint device activity and data detection in mMTC with single-phase non- coherent scheme as a hierarchical CS problem with two-level sparsity, where the device activity sparsity and transmitted pilot sequence sparsity are modeled as the system-level sparsity and the device-level sparsity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' We propose an AMP-based algorithm to solve the formulated CS problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' On this basis, we discuss the limitations of the AMP-based algorithm, which serves as the underlying motivation for designing the DL-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' We propose a DL-based algorithm, termed DL-mAMPnet, to conduct the device activity and data detection jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' DL-mAMPnet is composed of multiple AMP layers and one refinement module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The AMP layers are obtained by unfolding the AMP algorithm into a feedforward DNN, where trainable parameters are introduced to compensate for the inaccurate i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='d model of the traditional AMP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The refinement module exploits the unique spatial feature of the two-level sparsity structure to refine the output of the AMP layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In Section II, we present the system model and briefly introduce the non-coherent scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In Section III, we formulate a hierarchical CS problem with two-level sparsity and correspondingly derive an AMP-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In Section IV, we elaborate the structure of the proposed DL-mAMPnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In Section V, we present the parameter initialization and training method of the proposed DL-mAMPnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Simulation results are presented in Section VI, and conclusions are made in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Notations: We use normal lower-case, bold lower-case, and bold upper-case letters to denote scalars, vectors, and matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For matrix X, XT denotes its transpose, XH denotes its Hermitian transpose, |X| denotes its determinant, and ||X||F denotes its Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For vector x, ||x||p denotes its lp-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' E{·} denotes the expectation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' RM×N and CM×N denote the M × N dimensional real space and complex space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' CN(µ, Σ) denotes the multivariate complex Gaussian distribution with mean µ and covariance Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' SYSTEM MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Uplink Massive Access Scenario in mMTC Systems We consider a typical uplink massive access scenario in mMTC systems, where a set of randomly distributed single-antenna devices, denoted by N = {1, · · · , N}, communicate with 5 a BS equipped with M antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The uplink channel from device n to the BS is denoted by hn ∈ CM×1 and modeled as hn = � βngn, ∀n ∈ N, (1) where βn is the large-scale fading component and gn denotes the small-scale fading component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' We assume gn is distributed as CN(0, IM), and accordingly we have hn ∼ CN(0, βnIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' This paper adopts a block-fading channel model, where hn remains unchanged within channel coherence time but is independent from block to block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Due to the sporadic activity pattern of mMTC, only a small fraction of devices are active in each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' We assume that the devices are synchronized, and each device independently decides whether to access the channel with probability ϵ in each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Consequently, the device activity indicator for device n ∈ N is defined as αn = � � � � � 1, if device n is active, 0, otherwise, (2) where Pr(αn = 1) = ϵ and Pr(αn = 0) = 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' We further define the set of active devices within a block as K = {n ∈ N : αn = 1}, (3) and the number of active devices is K = |K|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The received signal y ∈ CM×1 at the BS is given by y = � n∈N αnhnxn + n = � k∈K hkxk + n, (4) where xn ∈ C is the transmitted signal of device n, and n ∈ CM×1 is the additive white Gaussian noise (AWGN) distributed as CN(0, σ2IM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' One-Phase Non-Coherent Scheme To successfully transmit the messages of the active devices, two schemes have been proposed in the literature, namely the two-phase coherent scheme and the one-phase non-coherent scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The two-phase coherent scheme divides each coherence block into two contiguous phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In the first phase, the active devices send their pilot sequences to the BS synchronously, and the BS jointly detects the device activity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', αn, as well as their corresponding channels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', hn, ∀n ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In the second phase, the active devices send their messages to the BS using the 6 remaining coherence block, and the BS decodes these messages based on the knowledge of device activity and channels obtained in the first phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Unlike the two-phase coherent scheme, the one-phase non-coherent scheme considered in this paper can jointly detect the active devices and the corresponding messages without explicit chan- nel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Specifically, in the non-coherent scheme, the transmitted messages are embedded in the index of the transmitted pilot sequence of each active device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' To this end, each device maintains a unique set of pre-assigned Q = 2J pilot sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' When a device is active, it sends a J-bit message by transmitting one sequence from the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' By detecting which sequences are received, the BS acquires both the identity of the active devices as well as the J-bit message from each of the active devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' We define the pilot sequences allocated for device n as: Sn = {s1 n, s2 n, · · · , sQ n }, (5) where sq n = [sq n1, sq n2, · · · , sq nL]T ∈ CL×1, 1 ≤ q ≤ Q, and L is the sequence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Note that the total number of pilot sequences is usually much larger than the length of pilot sequence (or the length of a coherence block), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', NQ ≫ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' As such, it is impossible to assign mutually orthogonal sequences to all devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Following the pioneering work [25], we adopt the random Gaussian sequences in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Specifically, each entry of the pilot sequences is generated from i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='d complex Gaussian distribution with zero mean and variance 1/L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', sq nl ∼ CN(0, 1/L), so that each pilot sequence has a unit norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', ||sq n||2 = 1, ∀n ∈ N and q = 1, · · · , Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For transmission, each active device selects exactly only one sequence from Sn based on its message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Then, the composite received signal Y ∈ CL×M of the non-coherent scheme can be expressed as Y = N � n=1 Q � q=1 αq nsq nhT n + N = N � n=1 SnXn + N, (6) where Xn = [α1 nhn, α2 nhn, · · · , αQ n hn]T ∈ CQ×M and αq n ∈ {0, 1} indicates whether or not sequence q of device n is transmitted, with a slight abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Recall that each device is active with probability ϵ, we have Q � q=1 αq n = � � � � � 1, with probability ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 0, with probability 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (7) By further concatenating all sequences of N devices as S = [S1, S2, · · · , SN] ∈ CL×NQ, the received signal in (6) can be simplified as Y = SX + N, (8) 7 where X = [XT 1 , XT 2 , · · · , XT N]T ∈ CNQ×M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The pictorial form of (8) is sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 1, which intuitively shows that X has a hierarchical sparse structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The hierarchical sparse structure comprises two levels of sparsity, including the system-level sparsity and the device- level sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The system-level sparsity means that most rows in X are zero, which is due to the sporadic traffic pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The device-level sparsity enforces that there is at most one non-zero row exists in Xn, ∀n, because each active device only transmits one pilot sequence from its pilot set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' = + Device-level Sparsity System-level Sparsity At most one non-zero row exists in each Most rows in are zero Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Pictorial form of the signal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' PROBLEM FORMULATION AND AMP-BASED JOINT DETECTION ALGORITHM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Problem Formulation Our goal is to detect the binary variable αq n that indicates both the activity of device n and its transmitted message, which can be achieved by recovering X from the received signal Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Once X is recovered, αq n can be determined by the rows of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Due to the hierarchical sparse structure of X, such problem is a classic CS problem with known measurement matrix S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Therefore, we can formulate the problem as follows: P1 : min X ||Y − SX||2 F (9) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' N � n=1 Q � q=1 I(Xnq,:) ≤ K, (10) Q � q=1 I(Xnq,:) ≤ 1, ∀n, (11) 8 where Xnq,: is the qth row of Xn and I(·) is the indicator function defined as I(x) = � � � � � 1, if x has non-zero elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (12) The constraint (10) comes from the system-level sparsity and the constraint (11) ensures the device-level sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' However, it is challenging to solve P1 directly due to the non-smooth constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Hence, we relax (11) into a l2,1-norm regularized least-square problem by replacing the indicator function with l2 norm as [26] min X 1 2||Y − SX||2 F + λ N � n=1 Q � q=1 ||Xnq,:||2, (13) where λ is the tunable parameter that balances the the sparsity of the solution and the mean square error (MSE) ||Y − SX||2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Although conventional CS algorithms such as orthogonal matching pursuit (OMP) and sparse Bayesian learning (SBL) can be directly used to solve (13), they suffer high computational complexity due to the matrix inverse operation, especially in mMTC system with massive devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In view of this, this paper utilizes the computationally efficient AMP algorithm as the main technique [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Review of the AMP Algorithm AMP refers to a class of efficient algorithms for statistical estimation in high-dimensional problems such as linear regression and low-rank matrix estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The goal of the AMP algorithm is to obtain an estimate of X with the minimum MSE based on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Starting with X0 = 0 and R0 = Y , the AMP algorithm can be described as follows: Xt+1,n = ηt,n(SH n Rt + Xt,n), ∀n, (14) Rt+1 = Y − SXt+1 + btRt, (15) where t = 0, 1, · · · is the index of the iteration, ηt,n(·) is the shrinkage function for device n that shrinks some items of its input to zero, and Rt is the corresponding residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The residual in (15) is updated with the “Onsager correction” term btRt, which substantially improves the performance of the AMP algorithm [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Note that ηt,n(·) is assumed to be Lipschitz-continuous and bt can be written as bt = 1 L N � n=1 η ′ t,n(SH n Rt + Xt,n), (16) 9 where η ′ t,n(·) is the first-order derivative of ηt,n(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In addition to improving the performance, the Onsager correction also enables the AMP algorithm to be analyzed by a set of state evolution equations in the asymptotic regime [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The asymptotic regime is when L, N → ∞, while their ratio converges to a positive constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', N/L → ρ where ρ ∈ (0, ∞), and while keeping the data length J fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' To facilitate the theoretical analysis, this paper considers a certain asymptotic regime where N → ∞, and the empirical distribution of the large-scale fading components βn’s converges to a fixed distribution pβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Define β ∼ pβ and Xβ ∈ CQ×M as a random matrix distributed as (1 − ϵ Q) �Q i=1 δxβ,i + ϵ Q �Q i=1 Phβ � j̸=i δxβ,j, where δxβ,i is the Dirac delta at zero corresponding to the element xβ,i and Phβ denotes the distribution hβ ∼ CN(0, βIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The state evolution equations can be written as the following recursions for t ≥ 0 [29] Σ0 = σ2IM + ρEβ{XH β Xβ}, (17) Σt+1 = σ2IM + ρEβ{(ηt(Xβ + V Σ 1 2 t ) − Xβ)H(ηt(Xβ + V Σ 1 2 t ) − Xβ)}, (18) where V ∈ CQ×M is a random matrix independent with Xβ, of which the rows are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' and each follows the distribution CN(0, IM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' It can be observed from (14) and (18) that applying ηt,n(·) to SH n Rt + Xt,n is statistically equivalent to applying ηt,n(·) to Xt,n + V Σ 1 2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Therefore, the input to the shrinkage function ηt,n(·) can be modeled as an AWGN-corrupted signal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', Zt,n = Xt,n + SH n Rt = Xt,n + V Σ 1 2 t , (19) In this case, the update given by (14) is statistically equivalent to a denosing problem, and thus ηt(·) can also be called “denoiser”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Hereafter, we use “shrinkage function” and “denoiser” interchangeably for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' AMP-Based Joint Device Activity and Date Detection Algorithm The core idea behind the joint detection algorithm is to first estimate X from Y , based on which αq n is determined according to the norm of each rows in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' To this end, we first derive the denoiser ηt,n(·) under the MMSE-optimal criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' After that, we observe that ηt,n(·) exhibits an asymptotic property, which motivates us to design a threshold-based strategy to extract αq n from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 10 1) Derivation of ηt,n(·): For notational simplicity, we omit the iteration index t in the fol- lowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' According to (19), the likelihood of Zn given Xn takes the form of PZn|Xn = Q � q=1 exp(−(zq n − xq n)HΣ−1(zq n − xq n)) πM|Σ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (20) Accordingly, the MMSE-optimal denoiser is given by the conditional expectation E{Xn|Zn} and can be expressed as ηn(Zn) = E{Xn|Zn} = [φ1 nΩnz1 n, · · · , φQ n ΩnzQ n ], (21) where Ωn = βn(βnIM + Σ)−1, (22) φq n = 1 1 + Q−ϵ ϵ exp(M(ψn − πq n)), (23) ψn = log(|IM + βnΣ−1|) M , (24) and πq n = zq n H(Σ−1 − (Σ + βnIM)−1)zq n M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (25) Proof: Please refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' It is important to realize that the MMSE-optimal denoiser ηn(·) is rather complicated as it involves the computation of the state evolution matrix Σ, where the matrix multiplication and expectation are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Hence, we simplify ηn(·) by using the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Theorem 1: Considering the asymptotic regime where both the number of devices N and the length of the pilot sequences L go to the infinity with their ratio converging to some fixed positive values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', N/L → ρ where ρ ∈ (0, ∞), the state evolution matrix Σt always remains as a diagonal matrix with identical diagonal entries after each iteration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', Σt = τ 2 t IM, ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (26) Correspondingly, the signal model given in (19) reduces to Zt,n = Xt,n + SH n Rt = Xt,n + τtV , (27) and the MMSE-optimal dnoiser given in (21)-(25) is simplified as ηn(Zn) = E{Xn|Zn} = [φ1 nωnz1 n, · · · , φQ n ωnzQ n ], (28) 11 where ωn = βn βn + τ 2, (29) φq n = 1 1 + Q−ϵ ϵ exp(M(ψn − πq n)), (30) ψn = log(1 + βn τ 2 ), (31) and πq n = βnzq n Hzq n τ 2(βn + τ 2)M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (32) Finally, τ 2 t can be obtained using the following recursions for t ≥ 0: τ 2 0 = σ2 + ρϵEβ{β}, (33) τ 2 t+1 = σ2 + ρ Q � q=1 Eβ{ φq ββτ 2 t β + τ 2 t } + ρ Q � q=1 Eβ{φq β(1 − φq β) β2zq n Hzq n (β + τ 2 t )2M }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (34) We omit the detailed proof here for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Interested readers can refer to theorem 1 in [8], where a similar derivation is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' It should be mentioned that the proposed Theorem 1 in this paper is essentially a generalization of Theorem 1 in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' When each device is assigned with only one pilot sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', Q = 1, the proposed Theorem 1 reduces to Theorem 1 in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 2) Threshold-Based Strategy: It can be seen from (28)-(30) that for large M, we have φq n → 1 if πq n > ψn and φq n → 0 if πq n < ψn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The asymptotic behavior of φq n indicates that it is reasonable to adopt a threshold-based strategy for solution refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Meanwhile, considering the device sparsity in (7), an element selection operation is necessitated to enforce all the elements except the one with the largest magnitude in each Xn to be zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Consequently, the proposed threshold- based strategy should be able to perform the following two operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Element Selection Operation: To surely guarantee the sparsity constraint in (11), we choose the largest row in each Xn = [x1 n, x2 n, · · · , xQ n ] and define the index of the largest element as i∗ n = arg max i xi n Hxi n, ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (35) Threshold-based Decisive Operation: After obtaining i∗ n, the binary variable vector αn = {α1 n, · · · , αQ n } can be given as αn = � � � ei∗n, if κi∗ n n > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 0, otherwise, (36) 12 where ei∗n is a one-hot vector of length Q with only the i∗ nth element equal 1 and the others equal 0, and the corresponding threshold is computed using (31) and (32) as κi∗ n n = zi∗ n n Hzi∗ n n βn τ 2 t (βn + τ 2 t )M − log � 1 + βn τ 2 t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (37) 3) Limitation: Although the traditional AMP-based algorithm can successfully recover aq n from Y , it has some inherent limitations: (i) The traditional AMP algorithm implicitly assumes Xn has a prior distribution with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' entries, which neglects the dependencies among the rows of Xn imposed by the device-level sparsity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (ii) The calculation of the denoiser ηt,n(·) and the threshold κn requires the exact value of βn, which is costly to obtain in a large-scale mMTC system with massive devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' AMP layer 1 AMP layer 2 AMP layer t AMP layer T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' AMP Layers Refinement Module … … Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Network architecture of the proposed DL-mAMPnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' DEEP LEARNING MODIFIED AMP NETWORK To address the aforementioned limitations, we propose a deep learning modified AMP network (DL-mAMPnet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The DL-mAMPnet is constructed by unfolding the AMP algorithm into a feedforward DNN, which inherits the mathematical model and structure of the AMP algorithm, thereby avoiding the requirements for accurate modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' On this basis, we introduce a few trainable parameters into the DL-mAMPnet to learn the active probability and the large-scale fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' By making the active probability trainable, we compensate for the inaccuracy caused by the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' assumption in the traditional AMP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' By making the large-scale fading coefficient trainable, we bypass the statistical measurements for the large-scale fadings of mas- sive devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' According to the threshold-based strategy in Section III-C, we further design a refinement module to guarantee the device-level sparsity and obtain the desired aq n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 13 As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 2, the proposed DL-mAMPnet consists of T uniform AMP layers and one refinement module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For the sake of clarity, each part of the DL-mAMPnet is elaborated respectively in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Input and Output To facilitate the learning process of DL-mAMPnet, the complex matrices need to be converted into the real domain and then vectorized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' To do this, we first express (8) as � � ℜ(Y ) ℑ(Y ) � � = � � ℜ(S) −ℑ(S) ℑ(S) ℜ(S) � � � � ℜ(X) ℑ(X) � � + � � ℜ(N) ℑ(N) � � , (38) where ℜ(·) and ℑ(·) denote the real and imaginary parts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The real and imaginary parts are then concatenated together and vectorized as ˜Y = vec([ℜ(Y )T, ℑ(Y )T]T) ∈ R2LM×1, (39) ˜S = � [ℜ(S), −ℑ(S)]T, [ℑ(S), ℜ(S)]T�T ⊗ IM ∈ R2LM×2NQM, (40) ˜ X = vec([ℜ(X)T, ℑ(X)T]T) ∈ R2NQM×1, (41) ˜ N = vec([ℜ(N)T, ℑ(N)T]T) ∈ R2LM×1, (42) where vec(·) is the vectorize operation that flattens a matrix into a vector in the order of columns, and ⊗ is the Kronecker product operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Consequently, (8) can be rewritten as ˜Y = ˜S ˜ X + ˜ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (43) According to the recursive formula in (14)-(15), the input to the DL-mAMPnet is chosen to be the the received signal, the estimated signal, and the residual, which are initialized as ˜ X0 = 0 and ˜R0 = ˜Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Meanwhile, unlike the existing AMP-inspired network that uses ˜ X [30], we adopt α = [α1 1, · · · , αQ 1 , α1 2, · · · , αQ N]T ∈ {0, 1}NQ×1 as the output of DL-mAMPnet, such that αq n can be directly obtained once DL-mAMPnet is well-trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 14 × × × × Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Detailed structure of the tth AMP layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' AMP Layer Since each layer has the same structure, we focus on the tth AMP layer of the DL-mAMPnet, of which the detailed structure is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Define the input as ˜ Xt−1, ˜Rt−1 and the output as ˜ Xt, ˜Rt, the tth AMP layer proceeds as follows ˜ Xt = ηt( ˜ Xt−1 + Bt ˜Rt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Θt), (44) ˜Rt = ˜Y − At ˜ Xt + ˜Rt−1 LM 2NQM � j=1 [ηt( ˜ Xt−1 + Bt ˜Rt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Θt)] ′ j, (45) where At and Bt are trainable matrices that acts as the matched filter and Θt = {θt,1, θt,2} is the trainable parameter set of ηt(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' It should be mentioned that the denoiser in (28)-(32) cannot be applied in the AMP layer, as the complex-to-real transformation and vectorization in (39)-(43) have changed the dimension and distribution of the corresponding matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Following the same derivation in Appendix A but considering ˜ X as a real-valued Bernoulli Gaussian variable and changing the dimension, ηt(·) in (28) can be expressed as [ηt( ˜Z)]j = β ˜Zj (β + τ 2 t ) � 1 + Q−ϵ ϵ exp(log(1 + β τ 2 t )1/2 − ˜ Z2 j β 2(β+τ 2 t )τ 2 t ) �, = ˜Zj (1 + τ 2 t β ) � 1 + � 1 + β τ 2 t exp(log( Q−ϵ ϵ ) − ˜ Z2 j 2(τ 2 t +τ 4 t /β)) �, (46) where ˜Zj is the jth element of ˜Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' As discussed in Section II-D, ηt(·) exploits an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' assumption that fails to effectively explore the correlated sparsity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' To tackle this issue, we replace log(Q−ϵ ϵ ) with a trainable parameter θt,1 = [θt,1,1, · · · , θt,1,2NQM]T ∈ R2NQM×1, such that the correlation among entries of ˜ X can be 15 learned and approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Meanwhile, to circumvent the need for the prior information of the large-scale fading, we introduce a trainable parameter θt,2 = [θt,2,1, · · · , θt,2,2NQM]T ∈ R2NQM×1 and substitute it for β in (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The trainable ηt(·) can then be defined as [ηt( ˜Z)]j = ˜Zj (1 + τ 2 t θt,2,j ) � 1 + � 1 + θt,2,j τ 2 t exp(θt,1,j − ˜ Z2 j 2(τ 2 t +τ 4 t /θt,2,j)) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (47) The derivative of ηt(·) is thus be given by [ηt( ˜Z)] ′ j = [ηt( ˜Z)]j ∂ ˜Zj = 1 + � 1 + θt,2,j τ 2 t exp(θt,1,j − ˜ Z2 j 2(τ 2 t +τ 4 t /θt,2,j))(1 + ˜ Z2 j (τ 2 t +τ 4 t /θt,2,j)) (1 + τ 2 t θt,2,j ) � 1 + � 1 + θt,2,j τ 2 t exp(θt,1,j − ˜ Z2 j 2(τ 2 t +τ 4 t /θt,2,j)) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (48) Note that to evade the computation of the expectation involved in τ 2, this paper adopts an empirical result where τ 2 is estimated by the standard deviation of the corrupted noise in ˜Z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', τ 2 t = || ˜Rt||2/ √ 2LM [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Remark 1: It is worth noting that the denoiser derived in (28) operates in a section-wise manner, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', acts on Q rows of each Xn, while the ηt(·) in the AMP layer operates row-by-row on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Although the section-wise manner may exploit the correlations better than the row-wise manner, it is quite challenging to be implemented in DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' This is because to realize such section-wise manner, we have to either construct N sublayers or impose N iterations in each AMP layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The former will heavily expand the network size and trainable parameters, reducing the scalability and stunting the training process of the DL-mAMPnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The latter will greatly increase the computational complexity of the DL-mAMPnet and negate the “deep unfolding” advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' It should also be noted that although the AMP layer can explore the correlated sparsity pattern with the help of trainable parameters, the device-level sparsity constraint in (7) is not surely guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Motivated by this consideration, we propose a felicitous method in the refinement module that utilizes the Maxpool-MaxUnpool operation to ensure device-level sparsity, as detailed in the subsection below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Refinement Module The refinement module should be capable of ensuring the device-level sparsity while extracting aq n from ˜ XT without explicit channel state information (CSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' To fulfil these functionalities, two components are integrated in the refinement module, namely the soft-thresholding denois- ing component and the hard-thresholding decision component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The soft-thresholding denoising component is intended to further denoise ˜ XT by exploiting the hierarchical sparse structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The 16 × 2NQ× M 2NQM × 1 2NQ × 1 2NQ × 1 2NQ × 1 2NQ × 1 2NQ × 1 2NQ × 1 2NQ × 1 2N × 1 2NQ × 1 2NQM × 1 1 × 1 (2NQM+1) × 1 2NQ × 1 NQ × 1 Reshape Absolute Conv FC+ReLU FC+ReLU FC+ReLU Sigmoid Average ReLU MaxPool Concatenate MaxUnpool FC+ Soft-thresholding Denosing Hard-thresholding Decision Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Detailed architecture of the proposed refinement module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' hard-thresholding decision component is aimed at implementing the threshold-based strategy in (35)-(37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The detailed structure of the refinement module is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 4 and elaborated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Soft-Thresholding Denoising: As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 1, the two-level sparsity exhibits a unique spatial structure that has not been utilized in the AMP layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Here, the soft-thresholding denoising aims to distill ˜ XT using such spatial feature, enhancing useful information while removing noise information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' To do this, we first de-vectorize ˜ XT and take the absolute value as X = |Vec−1( ˜ XT)| = [|ℜ(X)T|, |ℑ(X)T|]T ∈ R+2NQ×M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (49) Then, a convolutional layer with 1 × M kernel size is applied to X to combine the information from all M antennas and extract a coarse estimation of aq n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' This arrangement is motivated by the fact that all M elements in each row of X share the same aq n, as observed from (6) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The coarse estimation can be expressed as fθc(X), where fθc(·) is the function expression of the convolutional layer with parameter θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' After that, an average pooling with 1 × M kernel size is applied to X to get a 1-D average vector over M antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The 1-D vector ι = 1 M �M m=1 X:,m is forwarded into a two-layer fully-connected (FC) network to obtain a scaling parameter, such that the inner features of the average value among the 2NQ rows of X can be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The scaling parameter is then scaled to the range of (0, 1) using a sigmoid function, which can be written as follows ϑ = 1 1 + e −fθF C1 (ι), (50) 17 where ϑ is the scaling vector and fθF C1(·) is the function expression of the two-layer FC network with parameter θFC1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Next, ϑ is multiplied by ι to get the threshold as κST = ϑ ⊙ ι, (51) where ⊙ is the Hadamard product operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' This operation is inspired by the fact that the threshold for soft thresholding must be positive and not too large [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' If the threshold is larger than the largest value of fθc(X), then the output of soft thresholding will all be zeros, and thus the useful information will be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Finally, the obtained threshold κST is subtracted by fθc(X) and fed into a ReLU activation function as o = max(0, fθc(X) − κST), (52) where o denotes the output of the soft-thresholding denoising component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' We can observe from (52) that by keeping κST in a reasonable range, the useful information can be preserved while the noise information is eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' It is worth noting that, rather than being manually set by experts, such a threshold can be learned automatically in the proposed soft-thresholding denoising component, removing the need for the expertise of signal processing and the statistical characteristic of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 3 7 1 5 9 8 5 9 8 0 0 0 5 9 8 Pooling Indices MaxPool MaxUnpool Filter Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Illustration of the MaxPool-MaxUnpool process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Hard-thresholding Decision: It is challenging to directly implement the threshold-based strategy in DNNs, as (35) is non-differentiable and will stunt the backpropagation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' To tackle this issue, the hard-thresholding decision component elegantly uses the Maxpool and MaxUnpool procedures to ensure the device-level sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Maxpool is a down-sampling technique that uses a max filter to non-overlapping subregions of the initial input [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For each region represented by the filter, we will take the max of that region and create a new output 18 matrix where each element is the max of a region in the original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Maxunpool, in contrast, expands the output of the maxpool operation to its original size by upsampling and padding with zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Except for the maximum position, all the rest elements in the unpooled matrix are supplemented with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For an intuitive explanation, we illustrate the process of Maxpool and MaxUnpool in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' It can be observed from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 5 that in each filter, except for the largest value that remains unchanged, all the rest elements become 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Such manipulation perfectly executes the element selection operation in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' By setting the filter size as Q × 1, we enforce that at most one non-zero row exists in the Q rows of Xn, and therefore the device-level sparsity constraint in (11) can be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' It should also be mentioned that the pooling procedure is only a module that alters the dimension size during the deep learning process, which has no parameters and thus has no impact on network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' After guaranteeing the device-level sparsity, the onus shifts to performing the threshold-based decisive operation in (36), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', determining the binary sequence α by comparing the threshold κi∗ n n with the matrix obtained from the maxpool-maxunpool procedure Mp(Mup(o)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' However, some issues exist when determining α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The first issue is that the threshold in (37) may not be precise sufficiently because it is derived under an mismatched i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' To tackle this issue, we look afresh at (37) and find that the threshold is a function of β and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Since β has been represented by θ2 in (47), we concatenate θT,2 and τ 2 T outputted from the last AMP layer and feed it into an FC layer with ReLU activation function to learn the accurate threshold, which is denoted by κHT = max(0, fθF C2(θT,2, τ 2 T)), (53) where fθF C2 is the function expression of the FC network with parameter θFC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Then, the learned threshold κHT is subtracted by Mp(Mup(o)) and forwarded into an FC layer with parameter θFC3 to fulfil the threshold-based decisive operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The FC layer here has two functionalities: compressing the dimension from 2NQ × 1 to NQ × 1 and converting the κHT-Mp(Mup(o)) difference into a binary sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Mathematically, the optimal function for threshold-based binary decision is the signum function denoted as sng(x) = � � � 1, x > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 0, x ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (54) 19 However, since sng(x) is non-differentiable, it cannot be used in DNN, necessitating the devel- opment of a substitute function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 10 8 6 4 2 0 2 4 6 8 10 Input 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='9 1 Output Region with positive output and negative input Sgn (Optimal) Sigmoid Proposed (m=1) Proposed (m=5) Proposed (m=10) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The curves of the optimal signum, sigmoid, and hard-thresholding decision functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' When it comes to DL-based binary decisions, the sigmoid function is a popular choice and has been widely used in the literature [33], as it can map the input to the interval within [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The sigmoid function, nevertheless, is still inapplicable to the hard-thresholding decision module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The reasons are as follows: (i) The sigmoid function returns a continuous value between 0 and 1, implying that a threshold is further required to distinguish the outputted value as 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' However, it is usually non-trivial to design an appropriate threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (ii) According to (36), the output of the threshold-based decision should be strictly 0 with negative input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' However, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 6, there is a region where the output is still positive with negative input in the sigmoid function, which may introduce additional errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' To solve the above issues, we devise a novel hard-thresholding decision function, whose core idea is to cascade the ReLU function with tahn function and introduce a multiplier ϱ to approximate the cascaded function as a signum function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The proposed hard-thresholding decision function is given by fϱ(x) = max(0, eϱx − e−ϱx eϱx + e−ϱx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (55) By cascading the ReLU function with tahn function, we not only ensure that the output of the threshold-based decision is strictly 0 with negative input, but also guarantee the output with positive input approximates to 1 with the increment of ϱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The optimal signum, sigmoid, and hard-thresholding decision functions are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The figure shows that with the increase of ϱ, fϱ(·) gradually approximates to sng(x), validating the rationality of the proposed hard-thresholding decision function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 20 Remark 2: Although we restrict the application of the hard-thresholding decision component to the non-coherent transmission in mMTC, the proposed component can be used in any other scenarios where the signal has a special sparsity structure, such as the spatial modulation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Meanwhile, the devised hard-thresholding decision function can also be used in any bit-level detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' That is, the hard-thresholding decision component is a plug-and-play module with a wide range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' THE IMPLEMENTATION OF DL-MAMPNET A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Parameter Initialization In deep learning, parameter initialization plays a critical role in speeding up convergence and achieving lower error rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Choosing proper initialization values is especially important for the proposed DL-mAMPnet, as the DL-mAMPnet is built on the AMP algorithm and thus should preserve some essential features to ensure performance and interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' There are mainly three items needed to be considered for parametrization: the trainable matrices At and Bt, the denoiser parameter set Θt, and the refinement module parameters θRM = {θFC1, θFC2, θFC3, θC}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 1) Initializing At and Bt: It can be observed from (44)-(45) that the DL-mAMPnet imple- ments a generalization of the AMP algorithm in (14)-(15), wherein the matched filters (S, SH n ) manifest as (At, Bt) at iteration t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' However, such generalization does not enforce Bt = AH t and thus may not preserve the independent-Gaussian nature of the denoiser input (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' According to the analysis in [30], the desired nature maintains when At = υtS with υt > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Therefore, At is parameterized as υtS and (44)-(45) can be rewritten as ˜ Xt = υtηt( ˜ Xt−1 + Bt ˜Rt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Θt), (56) ˜Rt = ˜Y − S ˜ Xt + υt ˜Rt−1 LM 2NQM � j=1 [ηt( ˜ Xt−1 + Bt ˜Rt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Θt)] ′ j, (57) the derivation of which can be found in [30] and is omitted here for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In this paper, we initialize Bt = ˜ST and υt = 1, since such initialization can greatly expedite the convergence of the training process [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 2) Initializing Θt: For θ1, we initialize each element as log(Q−ϵ ϵ ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', initialize that each pilot sequence has the same active probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' This is because we have no prior information about the device activity and the transmitted pilot sequence index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' By adopting such a uniform initialization, the initial θ1 will have the minimum Euclidean distance from the actual value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 21 For example, consider a device with a 2-bit message and active indicator {1, 0, 0, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' If we start with a mismatched one-hot vector, then the Euclidean distance will be √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' If we initialize αn as {1 4, 1 4, 1 4, 1 4}, then the Euclidean distance will be � 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Therefore, the uniform initialization can accelerate the convergence as a shorter Euclidean distance may lead to faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The initial value of θ2 can be computed from the received signal strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Recall that each pilot sequence has a unit norm and hn ∼ CN(0, βnIM), each element of the initial θ2 is roughly given by || ˜Y ||2 2/ √ 2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 3) Initializing θRM: For all parameters in the refinement module, we adopt the He initial- ization [34] as it has been mathematically proved to be the best weight initialization strategy for the ReLU activation function [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Parameter Training 1) Training Algorithm: Aside from the network structure and parameter initialization, the training algorithm also determines the performance of the DL-mAMPnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The standard training strategy is the end-to-end training where all the parameters are optimized simultaneously by following the back-propagation rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' However, the end-to-end training is not appropriate for the DL-mAMPnet due to the following reasons: (i) The AMP algorithm aims to provide an estimate ˆ X(Y ) based on Y that minimizes the MSE EXY || ˆ X(Y )−X||2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' If the DL-mAMPnet is trained to learn the direct mapping from Y to α, the MSE optimality of the AMP layers may not be achieved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (ii) Even if the AMP layers and the refinement module are trained separately, the AMP layers can still easily converge to a bad local optimal solution due to overfitting [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For these reasons, we propose a layer-wise training strategy, the idea behind which is to decouple the training of each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The details are given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' There are totally T + 2 phases in the layer-wise training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In the first phase, we train the learnable parameters of the first AMP layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Then in the t phase, we train the first t AMP layers with the parameters of the first t − 1 AMP layers fixed as the parameters learned by the first t − 1 phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In the T + 1 phase, we train the whole network with only the parameters of the refinement module is learnable, while the parameters of the AMP layers are fixed as the parameters learned by the first T phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Finally, in the last phase, all the parameters are initialized as the parameters learned during the first T + 1 phases and then trained jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 22 Algorithm 1 Parameter training of the DL-mAMPnet via layer-wise training strategy Input: Training dataset DAMP, DRM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Output: Trained parameter {υt, Bt, Θt}T t=1 and θRM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Initialize parameters according to Section IV-B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' for t = 1 to T do Learn {υt, Bt, Θt}t with fixed {υt, Bt, Θt}t−1 t=1 based on the loss function (58);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' end for Learn θRM with fixed {υt, Bt, Θt}T t=1 based on the loss function (59);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Re-learn {υt, Bt, Θt}T t=1 and θRM based on the loss function (59);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' return {υt, Bt, Θt}T t=1 and θRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The training dataset DAMP for the first T phases comprises 100, 000 pairs of ˜ X and ˜Y , and the corresponding loss function is the MSE loss Lt( ˜Y ) = || ˜ Xt( ˜Y ) − ˜ X||2 2, t = [1, · · · , T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (58) The training dataset DRM for the last 2 phases has 100, 000 pairs of α and ˜Y , and the loss function is the binary cross entropy loss Lt( ˜Y ) = 1 NQ NQ � i=1 � α( ˜Y )i log αi + (1 − α( ˜Y )i) log(1 − αi) � , t = [T + 1, T + 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (59) The DL-mAMPnet is trained epoch by epoch with the training dataset using the Adam optimizer, while within an epoch, the whole training dataset is shuffled and split into batches with the size of 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='1 2) Training Dataset: The training dataset is synthetically generated as follows: (i) Generating αn: K active devices are randomly selected among N devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Then, each active device is randomly assigned with a Q-dimensional one-hot vector, and each inactive device is assigned with a Q-dimensional zero vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (ii) Generating Xn: The uplink channel of device n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', hn, is first generated according to (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Then Xn is obtained by multiplying hn and αn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (iii) Generating Y : The pilot sequence Sn is generated by sampling from complex Gaussian distribution with zero mean and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Given Xn and Sn, Y can be directly obtained according to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 1It should be mentioned that the number of epochs and the learning rate are different for each phase, which are empirically determined in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 23 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' SIMULATION RESULTS In this section, extensive simulations are provided to verify the effectiveness of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The setup is as follows unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' We consider a mMTC system with N = 100 devices for illustration purpose, although the proposed algorithm can be used for a much larger-scale system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Each device accesses the BS independently with probability ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='1 at each coherence block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The large-scale fading coefficient for device n is βn = 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='1 − 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='7 log10(dn) in dB, where dn is the distance between device n and the BS that follows a uniform distribution within [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='05, 1] km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The small-scale fading coefficient for each device follows the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' multivariate complex Gaussian distribution with zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The power spectral density of the AWGN at the BS is assumed to be −169 dBm/Hz [8] and the bandwidth of the wireless channel is 1 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The number of AMP layers in the DL-mAMPnet is set to be T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The training epochs and learning rate for each training phase are set to be {2, 000, 1, 500, 1, 000, 1, 000, 1, 500, 5, 000} and {2 × 10−5, 2 × 10−5, 2 × 10−5, 2 × 10−5, 1 × 10−5, 1 × 10−5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='2 We train the DL-mAMPnet with 80, 000 training samples and test with 20, 000 data samples, which are randomly drawn from DAMP for the first 4 phases and DRM for the last 2 phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The DL-mAMPnet is trained and tested by on an x86 PC with one Nvidia GeForce GTX 1080 Ti graphics card, and Pytorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='0 is employed as the backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The traditional AMP-based algorithm with TAMP = 50 iterations and the covariance-based method with TCov = 50 iterations [16] are employed as the benchmark and evaluated on the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' In addition, the SER is adopted as the performance metric: SER = 1 N �N n=1 I( ˆαn ̸= αn), where ˆαn and αn denote the estimated pilot sequence activity for device n and its ground truth, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Performance of the DL-mAMPnet Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 7 depicts the SER versus L with different values of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' It is observed that both the SER of the DL-mAMPnet and AMP-based algorithm decrease as L and M increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Although the SER of the covariance-based algorithm is lowest when L is small, it becomes saturated when L exceeds some point, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', L = 40 when M = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' This is mainly due to the suboptimality of the 2All the parameters are empirically determined using the general workflow, where the training starts with relatively small values and increases the values until the learning performance cannot be further improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 24 10 20 30 40 50 60 70 80 90 100 Pilot Sequence Length: L 10-4 10-3 10-2 10-1 100 SER AMP, M=8 AMP, M=16 AMP, M=32 Covariance, M=8 Covariance, M=16 Covariance, M=32 DL-mAMPnet, M=8 DL-mAMPnet, M=16 DL-mAMPnet, M=32 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' SER performance versus the pilot sequence length L for J = 1 bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 0 5 10 15 20 25 30 35 40 Number of Receiving Antennas: M 10-4 10-3 10-2 10-1 100 SER AMP, L=50 AMP, L=60 AMP, L=70 Covariance, L=50 Covariance, L=60 Covariance, L=70 DL-mAMPnet,L=50 DL-mAMPnet,L=60 DL-mAMPnet,L=70 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' SER performance versus the number of receiving antennas M for J = 1 bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' fixed threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='3 Meanwhile, the proposed DL-mAMPnet notably outperforms the AMP-based algorithm by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For example, the proposed DL-mAMPnet achieves more than 10 pilot length gain over the AMP-based algorithm when L is larger than 70, which indicates that the proposed DL-mAMPnet can reduce the required pilot sequence length, lowering the difficulty of pilot design and adapting to fast-changing channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Moreover, although for any M, the SERs of both the DL-mAMPnet and AMP-based algorithm decrease over L, the reduction is faster 3As observed from (37), the threshold is variable and related to system parameters such as signal power and receiving antenna numbers, whereas the covariance-based algorithm adopts a fixed threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Since there is no concrete method to design such a fixed threshold, we empirically set the threshold of the covariance-based algorithm to be βn/2 in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 25 10 20 30 40 50 60 70 80 90 100 Pilot Sequence Length: L 10-3 10-2 10-1 100 SER AMP, M=16,J=1bit AMP, M=16,J=2bits Covariance, M=16,J=1bits Covariance, M=16,J=2bits DL-mAMPnet, M=16,J=1bit DL-mAMPnet,M=16,J=2bits Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' SER performance versus the pilot sequence length L with different lengths of transmitted messages J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' when M is 32 as compared to that when M is 8, which shows that increasing the number of receiving antennas can further reduce the required pilot sequence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 8 shows the SER versus M for various values of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' We observe that for the DL-mAMPnet and AMP-based algorithm, the SER drops effectively as M increases, whereas for the covariance- based algorithm, there are error floors in the SER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Moreover, the DL-mAMPnet needs fewer receiving antennas to achieve the same performance as the AMP-based algorithm, implying that the proposed DL-mAMPnet can reduce demand for receiving antennas, resulting in lower deployment cost and energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 9 plots the SER versus L, with 2 different lengths of transmitted messages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', J = 1 bit and J = 2 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The number of receiving antennas is M = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' It can be seen that the SERs of all three algorithms increase as the length of transmitted messages increases, which implies that the performance of both algorithms deteriorates when more messages are transmitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' An important point is that as the message length increases, the performance gap between the proposed DL- mAMPnet and the other two algorithms increases, indicating the potential of the DL-mAMPnet to handle long packet size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Visualization of the DL-mAMPnet To offer more insights of the proposed DL-mAMPnet, we present the visualization of the outputs of each component of a well-trained DL-mAMPnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For clarity, we only present the case where N = 10 devices transmit 1-bit message with ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='1 active probability, L = 10 pilot 26 0123 0 4 8 12 16 20 24 28 32 36 (a) 0123 0 4 8 12 16 20 24 28 32 36 (b) 0 0 4 8 12 16 20 24 28 32 36 (c) 0 0 4 8 12 16 (d) 0 0 4 8 12 16 (e) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' A visualization of a well-trained DL-mAMPnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (a) ˜ XT , the output of AMP layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (b) The ground truth ˜ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (c) Mp(Mup(o)), the output of the maxpool- maxunpool procedure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (d) ˆα, the output of the refinement module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (e) The ground truth α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' sequence length, and M = 2 receiving antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For visualization, we transform the outputs of each component to the reverse grayscale images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Specifically, the elements of each output matrix are normalized to an interval within [0, 1], where 0 and 1 are represented by white color and black color, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' It should be mentioned that we take the absolute value of ˜ X and ˜ XT to show the signal strength difference more intuitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The output of the AMP layers and its ground truth are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 10(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 10(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' It can be seen that the non-zero rows of ˜ X are correctly recovered, paving the way for the subsequent refinement progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Then, the output of the maxpool-maxunpool procedure is visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 10(c), where the largest of the two adjacent rows is retained and the other becomes 0, demonstrating the validity of the maxpool-maxunpool procedure in ensuring the device-level sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 10(d) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 10(e) are the visualizations of ˆα and α, where we find that the pilot sequence activity is perfectly estimated by the well-trained DL-mAMPnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Moreover, it is observed from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 10(c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 10(e) that the pilot sequence activity is correctly reserved in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 10(c) (the 1st, 4th, 21st, and 24th rows), which indicates the effectiveness of the proposed soft-thresholding denoising component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Computational Complexity Analysis Finally, we analyze the computational complexities of the traditional AMP-based algorithm and DL-mAMPnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 27 For the traditional AMP-based algorithm, the computational complexity mainly comes from the matrix multiplication in (14)-(15) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Since SH n ∈ CQ×L, Rt ∈ CL×M, S ∈ CL×NQ, and Xt+1 ∈ CNQ×M, the computational complexity for N devices and TAMP iterations is O(4TAMP(NQLM + NQLM)) = O(8TAMPNQLM), where the proportional constant “4” appears because a complex multiplication requires 4 real multiplications, the former “NQLM” comes from the multiplication between SH n and Rt for N devices and the latter “NQLM” comes from the multiplication between S and Xt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' After the iterative process, the AMP-based algorithm requires the element selection operation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', (35)) whose computational complexity is O(4NQM), and the threshold calculation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', (37)) whose computational complexity is O(4NQM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Taking all the operations into account, the computational complexity of the AMP- based algorithm is given by O(8TAMPNQLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For the proposed DL-mAMPnet, we focus on the computational complexity of online imple- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The computational complexity of the AMP layers comes from the matrix multipli- cation Bt ˜Rt−1 and At ˜ Xt, which is O(8TDLNQLM 2) with TDL denoting the number of AMP layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For the refinement module, the computational complexity is mainly resulted from the FC and convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For a FC layer with Nl−1 input and N1 output, its computational complexity is given by O(Nl−1N1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For a convolutional layer with a H × W input and a Hf × Wf filter, its computational complexity can be expressed as O(HWHfWf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Therefore, the total computational complexity of the refinement module is O(4N 2Q2M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Consequently, the computational complexity of DL-mAMPnet is O(8TDLNQLM 2 + 4N 2Q2M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' From the above discussions, it seems that the proposed DL-mAMPnet can achieve better performance at the expense of a higher computational complexity compared to the AMP-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' However, as observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 7-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' 9, the DL-mAMPnet with TDL = 4 AMP layers outperforms the AMP-based algorithm with TAMP = 50 iterations, indicating that the proposed DL-mAMPnet may need less computational complexity to achieve the same SER performance with the AMP-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' CONCLUSION This paper has proposed a novel DL-based algorithm, termed DL-mAMPnet, for the joint device activity and data detection in mMTC with a single-phase non-coherent scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Trainable parameters have been added in the DL-mAMPnet to compensate for the inaccuracy caused by the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' assumption in the traditional AMP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' A refinement module has been further 28 designed to enhance the SER performance and guarantee the device-level sparsity by exploiting the correlated sparsity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' The proposed algorithm can be applied to scenarios where massive users intermittently transmit small packets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=', smart home and industrial control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' For the future work, we will investigate the pilot sequence design scheme to maintain orthogonality and mitigate the inter-device interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' APPENDIX A DERIVATION OF MMSE DENOISER (21) To enable the derivation of the conditional probability PXn|Zn, we assume xq n is independent with each other, and thus we have Pxq n = � 1 − ϵ Q � δ + ϵ Q exp(−xq n H(βnIM)−1xq n) πM|βnIM| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (60) According to (20), the likelihood of observing zq n given xq n is Pzq n|xq n = exp(−(zq n − xq n)HΣ−1(zq n − xq n)) πM|Σ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (61) Denoting k as the proportional constant, Pxq n|zq n can be computed using the Bayes’ formula as follows Pxq n|zq n = kPzq n|xq nPxq n = k � (1 − ϵ Q)δ + ϵ Q exp(−xq n H(βnIM)−1xq n) πM|βnIM| � �exp(−(zq n − xq n)HΣ−1(zq n − xq n)) πM|Σ| � = k � (1 − ϵ Q)exp(−zq n HΣ−1zq n) πM|Σ| δ + ϵ Q exp(−xq n H(βnIM)−1xq n − (zq n − xq n)HΣ−1(zq n − xq n)) π2M|βnIM||Σ| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (62) Note that xq n H(βnIM)−1xq n + (zq n − xq n)HΣ−1(zq n − xq n) = (xq n − ζ)HΞ−1(xq n − ζ) + zq n H∆−1zq n, where Ξ = ( 1 βnIM + Σ−1), ζ = ΞΣ−1zq n, and ∆ = βnIM + Σ, (62) can be rewritten as Pxq n|zq n = k � (1 − ϵ Q)exp(−zq n HΣ−1zq n) πM|Σ| δ + ϵ Q exp � −(xq n − ζ)HΞ−1(xq n − ζ) − zq n H∆−1zq n � π2M|βnIM||Σ| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (63) 29 Since � Pxq n|zq n dxq n = 1, k can be obtained by integrating (63) out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Accordingly, we have k = � (1 − ϵ Q)exp(−zq n HΣ−1zq n) πM|Σ| + ϵ Q exp � −zq n H∆−1zq n � |Ξ| πM|βnIM||Σ| �−1 (a) = � (1 − ϵ Q)exp(−zq n HΣ−1zq n) πM|Σ| + ϵ Q exp � −zq n H∆−1zq n � πM|∆| �−1 , (64) where (a) holds because | 1 βnIM + Σ−1| = |βnIM||Σ|/|βnIM + Σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' Substituting (64) into (62), Pxq n|zq n can be determined as Pxq n|zq n = e−(xq n−ζ)HΞ−1(xq n−ζ)ϵ|Σ| + (Q − ϵ)e−zq n H(Σ−1−∆−1)zq nπM|Ξ||∆|δ ϵπM|Ξ||Σ| + (Q − ϵ)e−zq n H(Σ−1−∆−1)zq nπM|Ξ||∆| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (65) Hence, the conditional expectation E{xq n|zq n} is given by E{xq n|zq n} = � xq nPxq n|zq n dxq n = ζϵ|Σ| ϵ|Σ| + (Q − ϵ)e−zq n H(Σ−1−∆−1)zq n|∆| = βn(βnIM + Σ)−1zq n 1 + Q−ϵ ϵ |IM + βnΣ−1|e−zq n H(Σ−1−(Σ+βnIM)−1)zq n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfo_hK/content/2301.00516v1.pdf'} +page_content=' (66) The MMSE-optimal denoiser in (21)-(25) can be straightforwardly obtained from (66) through simple mathematical transformation.' 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@@ -0,0 +1,1788 @@ +Draft version January 3, 2023 +Typeset using LATEX preprint style in AASTeX631 +Disk Wind Feedback from High-mass Protostars. II. The Evolutionary Sequence +Jan E. Staff,1 Kei E. I. Tanaka,2 Jon P. Ramsey,3 Yichen Zhang,3 and Jonathan C. Tan4 +1Department of Space, Earth & Environment, Chalmers University of Technology, Gothenburg, Sweden +and +College of Science and Math, University of the Virgin Islands, St Thomas, 00802, United States Virgin Islands +2Center for Astrophysics and Space Astronomy, Department of Astrophysical and Planetary Sciences, University of Colorado Boulder, +Boulder, CO 80309, USA +and +ALMA Project, National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan +3Department of Astronomy, University of Virginia, Charlottesville, VA 22904, USA +4Department of Space, Earth & Environment, Chalmers University of Technology, Gothenburg, Sweden +and +Department of Astronomy, University of Virginia, Charlottesville, VA 22904, USA +(Dated:) +ABSTRACT +Star formation is ubiquitously associated with the ejection of accretion-powered outflows that carve +bipolar cavities through the infalling envelope. This feedback is expected to be important for regulating +the efficiency of star formation from a natal pre-stellar core. These low-extinction outflow cavities +greatly affect the appearance of a protostar by allowing the escape of shorter wavelength photons. +Doppler-shifted CO line emission from outflows is also often the most prominent manifestation of +deeply embedded early-stage star formation. Here, we present 3D magneto-hydrodynamic simulations +of a disk wind outflow from a protostar forming from an initially 60 M⊙ core embedded in a high +pressure environment typical of massive star-forming regions. We simulate the growth of the protostar +from m∗ = 1 M⊙ to 26 M⊙ over a period of ∼100,000 years. The outflow quickly excavates a cavity +with half opening angle of ∼ 10◦ through the core. This angle remains relatively constant until the star +reaches 4 M⊙. It then grows steadily in time, reaching a value of ∼ 50◦ by the end of the simulation. +We estimate a lower limit to the star formation efficiency (SFE) of 0.43. However, accounting for +continued accretion from a massive disk and residual infall envelope, we estimate that the final SFE +may be as high as ∼ 0.7. We examine observable properties of the outflow, especially the evolution of +the cavity opening angle, total mass and momentum flux, and velocity distributions of the outflowing +gas, and compare with the massive protostars G35.20-0.74N and G339.88-1.26 observed by ALMA, +yielding constraints on their intrinsic properties. +1. INTRODUCTION +Low-mass stars and their associated accretion disks +form from gravitationally bound cores (Shu et al. 1987) +and are frequently associated with the launching of bipo- +lar jets and outflows (for reviews, see, e.g., Frank et al. +2014; Bally 2016). The magnetocentrifugal mechanism +(Blandford & Payne 1982; Pudritz & Norman 1983; +Konigl & Pudritz 2000) is widely thought to be respon- +Corresponding author: Jan E. Staff +jestaff.astro@gmail.com +sible for launching, accelerating and collimating these +protostellar outflows. In this scenario, the combination +of large-scale magnetic fields with gravity and rotation +results in the ejection, acceleration and then collima- +tion of gas originating from the surface of the accretion +disk. +A number of numerical simulation studies have +been performed to investigate this process across a va- +riety of different conditions and assumptions (e.g., Shi- +bata & Uchida 1985; Uchida & Shibata 1985; Ouyed +et al. 2003, 1997; Ouyed & Pudritz 1997; Romanova +et al. 1997; Krasnopolsky et al. 1999; Ramsey & Clarke +2011; Staff et al. 2010, 2015, 2019; Anderson et al. 2006; +Zanni et al. 2007; Te¸sileanu et al. 2012; Sheikhnezami +arXiv:2301.00749v1 [astro-ph.GA] 2 Jan 2023 + +2 +et al. 2012; Stute et al. 2014; Stepanovs & Fendt 2014, +2016; Ramsey & Clarke 2019; Gressel et al. 2020; Mat- +tia & Fendt 2020a,b). +However, alternative theoreti- +cal scenarios have also been proposed as being relevant +for outflow launching, including: the X-wind model in- +volving the interaction of the protostellar magnetic field +with the inner disk (e.g., Lovelace et al. 1991; Shu et al. +2000); stellar wind driven outflows (e.g., Matt & Pudritz +2005); and magnetic pressure driven outflows (Lynden- +Bell 1996). +Observationally, support for the disk wind model in +low- and intermediate-mass systems has been provided +by high angular resolution observations of a handful of +systems, e.g., TMC1A (Bjerkeli et al. 2016), HH212 (Lee +et al. 2017), DG Tau B (de Valon et al. 2020), and IRAS +21078+5211 (Moscadelli et al. 2022). In each case, the +launching of the outflow can be traced to the accretion +disk, demonstrating a launching radius that extends out +to scales of up to ∼ 20 au from the central protostar. +The formation of high-mass stars is more difficult to +characterize observationally as there are fewer sources, +they are farther away and they are more obscured by +surrounding gas and dust. Nevertheless, massive star +formation is also typically observed to be associated with +the launching of bipolar jets and outflows (see, e.g., Arce +et al. 2007; Tan et al. 2014; Beltr´an & de Wit 2016; +Hirota et al. 2017). +For example, the central source +powering HH 80 and HH 81 (IRAS 18162-2048) (Marti +et al. 1993), and G339.88-1.26 (Zhang et al. 2019) are +both associated with highly collimated outflows. +An- +other massive protostar, G35.20-0.74N, has also been +found to launch a highly collimated jet (e.g., Fedriani +et al. 2019). Indeed, Caratti o Garatti et al. (2015) found +that outflows from a number of intermediate and high- +mass protostars appear as scaled-up versions of those +from low-mass protostars, while Sandell et al. (2020) +also found this to be the case for the outflow from the +massive protostar NGC 7538 IRS1. Wider angle molec- +ular outflows have also been observed from massive pro- +tostars (e.g. Beuther et al. 2002; Wu et al. 2004; Zhang +et al. 2013, 2014a; Maud et al. 2015). In general, the +trend is that higher luminosity, i.e., more massive, pro- +tostars tend to have more powerful and more massive +outflows with wider opening angles than their low-mass +counterparts. +McKee & Tan (2002) suggested that a combination +of turbulence and magnetic pressure provides most of +the support in a massive pre-stellar core against gravity. +In this “Turbulent Core Accretion” (TCA) model, high- +mass star formation is a scaled-up version of low-mass +star formation, with accretion rates expected to be ∼ +10−4 to ∼ 10−3 M⊙ yr−1, compared to ∼ 10−6 to ∼ +10−5 M⊙ yr−1 in lower-mass cores. If that is the case, +then outflows from forming massive stars can therefore +also be a scaled-up version of the outflows from lower- +mass forming stars. +Other formation scenarios for high-mass stars have +also been proposed. Bonnell et al. (1998) suggested that +high-mass stars form by the collision of multiple smaller +objects that formed close together. +Another possibil- +ity suggested by Bonnell et al. (2001) is that massive +stars form together in the central region of dense proto- +clusters, where most of the mass is accreted from a glob- +ally collapsing clump (see Tan et al. 2014 for a review +of these scenarios). This could lead high-mass stars to +accrete from smaller disks that change orientation over +time, leading to outflows that also keep changing direc- +tions (Goddi et al. 2020). +In contrast to outflows from low-mass protostars, it +is still debated whether or not strong magnetization is +required to drive an outflow from high-mass protostars. +For example, Machida & Hosokawa (2020) found that, +in their simulations, the outflow launching failed or was +much delayed unless the initial cloud was strongly mag- +netized. In contrast, Beuther et al. (2020), based on ob- +servations, argued for a weak magnetization in the case +of G327.3, despite it also having an outflow. The direc- +tion of the magnetic field in the core is also debated; in +some cases, it has been found to be parallel to the out- +flow and perpendicular to the disk (Carrasco-Gonz´alez +et al. 2010; Sanna et al. 2015), while other studies have +found that the outflow axis is randomly oriented with +respect to the core-field (Zhang et al. 2014a). From an +analysis of about 200 outflows, Xu et al. +(2022) find +evidence for preferential alignment of outflow directions +with large-scale B−fields, but with significant scatter +for any given outflow to B−field to orientation. +Staff et al. (2019) (hereafter Paper I) presented 3D +magneto-hydrodynamic (MHD) simulations of disk wind +outflows from a 60 M⊙ core, but with the protostellar +mass, accretion rate and mass outflow rate held at fixed +values representing various stages of the protostellar evo- +lution. The method was to run each simulation for a +roughly a local accretion time to infer the properties of +the outflow cavity - envelope system. However, because +of this approximation this method involved significant +uncertainties. +In this paper, i.e., Paper II, we present similar MHD +simulations as Paper I, but now following the protostel- +lar evolutionary sequence consistently, i.e., as its mass +grows from m∗ = 1 M⊙ to more than 24 M⊙. As in +Paper I, we assume that a star is growing from a 60 M⊙ +core embedded in a clump with mass surface density of +Σcl = 1 g cm−2 within the framework of the Turbulent + +3 +Core Accretion model of McKee & Tan (2002, 2003). A +disk-wind (launched from the accretion disk) is injected +into the simulation box, where some envelope material +becomes entrained by the outflow. We simulate the out- +flow as it propagates through the envelope to investigate +the interaction between the wind and the envelope ma- +terial, and to investigate how much envelope material +is pushed away, providing us with an estimate of the +star formation efficiency. We also compare our simula- +tion results with observations of outflows from massive +protostars. +In §2 we describe our numerical methods. We present +our results in §3 and discuss their implications in §4. +Finally, we summarize our findings in §5. +2. METHODS +The goal of this work is to simulate a magnetically- +powered outflow from a massive, growing protostar. Us- +ing the ZEUS-MP code (Norman 2000), we conduct a +3D, ideal MHD simulation of an outflow from a massive +protostar in the framework of the turbulent core accre- +tion model (McKee & Tan 2003; Zhang et al. 2014b; +Zhang & Tan 2018). +As in Paper I, we consider an +initial core of mass of 60 M⊙. However, instead of sim- +ulating a sequence of separate models for different fixed +values of protostellar mass, m∗, here we follow the evo- +lution of a single simulation and the resulting outflow +for more than 100,000 years as the central star grows +from an small initial mass of m∗ = 1 M⊙. The setup of +the simulation is described below. +2.1. Simulation domain and boundary conditions +We use a Cartesian grid with 168 × 280 × 280 cells +in the x1, x2, and x3 directions, respectively, for our +“medium” resolution simulation. +A “high” resolution +simulation is also run for the earlier phases of the evolu- +tion with 336 × 560 × 560 cells. A logarithmic grid (“ra- +tioed” in ZEUS terminology) is employed, where cells +become larger in each direction in a regular fashion as +the distance from the origin increases. This allows us to +cover a fairly large spatial region, while maintaining a +reasonably high resolution in the central region. The x1 +direction (perpendicular to the disk and parallel to the +outflow) extends from 100 au above the disk midplane +to 26, 500 au, while the x2 and x3 directions (parallel to +the disk plane) extend out to ±16, 000 au. Compared +to the Paper I simulations, this domain is about twice +as long in the x1 direction, and slightly larger in the x2 +and x3 directions. +All boundaries, except for the inner x1 boundary, are +outflow boundaries. +The inner x1 boundary is more +complicated, as the outflow is injected through it, and +mass can “accrete” onto the disk through it. The fastest +part of the disk wind is injected in a circular region with +radius ri centered on the origin. Following Paper I, ri +is related to the size of the disk around the protostar, +rd (see eq. 2 of Paper I). Just outside of the injection +region is a smoothing region, through which material +is also injected. +The role of this smoothing region is +to gradually transition from the density and velocity of +the injected disk wind profile to that of the surround- +ing environment. The smoothing region has a radius of +ro = 1.8ri, somewhat larger than the value of ro = 1.3ri +used in Paper I to ensure that it contains several cells +in the x2 and x3 directions at all times. Going further +out is the accretion region, extending from ro to racc, +through which material is removed to join the accretion +disk at a controlled rate. Beyond this, we use reflecting +boundaries, to prevent any additional mass from flowing +off the grid. +The value of ri (and thus also ro) increases during the +evolution as the star grows in mass, since rd ∝ m2/3 +∗ +in +the fiducial model of Zhang et al. (2014b) in the limit +of constant star formation efficiency, a fixed disk to star +mass ratio, and a constant profile of rotational energy +to gravitational energy ratio of material in the initial +core. The radius of the accretion region, racc, adjusts +over time so that the integrated mass flow rate through +the annulus given by racc−ro that has outflow boundary +conditions is ˙msim = 1 +2 ˙m∗(1 + 1/3 + 1/10) ≃ 0.72 ˙m∗. +Note, the term 1/3 accounts for the growth of the accre- +tion disk, which is assumed to have a mass md = m∗/3). +The term 1/10 is present to account for the injected +mass flux of the disk wind that is immediately returned +to the simulation grid through the injection region. The +factor 1/2 is present since we simulate only one hemi- +sphere. The outer radius of the accretion region, racc, +is adjusted so that the desired accretion rate is achieved +via this region of outflow boundary condition. +2.2. Initial core +We initialize the simulation with a 1 M⊙ protostar +located at the origin of our coordinate system, which is +100 au below the inner x1 boundary. On the grid, we +include one hemisphere of a 60 M⊙ core, with a radius +of 12,000 au, which is the size expected for such a core +embedded in a clump with mass surface density of Σcl = +1 g cm−2. +In the TCA model, the fiducial initial density struc- +ture of the prestellar core is assumed to be spherical, +with a power-law of the form ρ ∝ r−kρ with kρ = 3/2. +Thus our density structure is given by +ρ(t = 0) = ρs (r/Rc)−3/2 , +(1) + +4 +where ρs is the density at the surface of the core. Note, +in Paper I, which was mainly considering snapshots of +later phases of the evolution, we adopted kρ = 1 as an +approximation of the expected structure that develops +in the expansion wave of the collapse solution. For our +core with kρ = 3/2, we have ρs = 2.5 × 10−18 g cm−3, +i.e., nH = 1.1 × 106 cm−3 assuming a mass per H of +2.34 × 10−24 g cm−3. Beyond Rc we adopt a constant +ambient density of 0.1ρs. The material in the core and +its surroundings is initialized to be at rest. +Following Paper I, the initial magnetic field configura- +tion is the canonical Blandford & Payne (“BP”) config- +uration (Blandford & Payne 1982), with a constant field +added to it to ensure that the core flux is ∼ 1 mG × R2 +c. +The BP configuration is a force-free, hour-glass shaped, +purely poloidal magnetic field configuration. +At the +mid-plane, the BP field varies as Bp ∝ r−1.25. +The 1D velocity dispersion of the fiducial 60 M⊙ +prestellar core, i.e., assuming virial equilibrium, is +1.09(Mc/60 M⊙)1/4(Σcl/1 g cm−2)1/4 km s−1. +In our +simulation we adopt an isothermal equation of state +with an effective sound speed, i.e., signal speed, of +cs = 0.90 km s−1. This choice is made so that the core +is moderately sub-virial and will undergo gravitational +contraction. +The gravitational field is treated with a simple ap- +proximation in which the mass of the star and the disk, +residing outside of the simulation domain, are treated +as a point mass. For the contribution of the potential +of the envelope material, we assume a simple model of +a fixed core envelope size, i.e., of radius Rc, and a fixed +power law index describing the radial distribution, i.e., +ρ ∝ r−3/2, but with the normalization of the profile +adjusted to match the mass that is remaining in the +envelope. +When the simulation starts, the core immediately be- +gins to contract as the initial setup is unstable to grav- +itational collapse. +Initially, the plasma-β (i.e., where +β ≡ Pgas/Pmag) is slightly above unity in the core. +However, as the envelope collapses, the plasma-β drops +below unity, meaning that the magnetic field starts to +dominate. The collapse will therefore not be spherically- +symmetric towards the protostar, but instead be guided +along the field lines towards the mid-plane. +2.3. Injection of the disk wind +We launch the disk wind through the injection re- +gion on the inner x1 boundary, with ˙minj = 1 +2 +1 +10 ˙m∗ = +0.05 ˙m∗. We also enforce that the injected outflow has +the same momentum rate in the x1 direction as in Zhang +et al. (2014b). Together, this can be used to constrain +the injected density and velocity in the x1 direction (per- +pendicular to the injection boundary). As in Paper I, +we then have an injected density: +ρinj = +� +� +� +� +� +exp (0.0289 rcyl/r∗)φρρ0 +rcyl < x0 +2.77 +�rcyl +x0 +�−1 +φρρ0 +rcyl ≥ x0 +(2) +and an injected v1 velocity: +vinj = (rcyl/r∗)−1/2φinjvK∗, +(3) +where r∗ is the stellar radius, x0 = 35.3r∗, rcyl is the +distance from the x1 axis, ρ0 is the injection density +at the axis, vK∗ is the Keplerian speed on the stellar +surface. φρ and φinj are time dependent dimensionless +factors that are needed in order to obtain the desired +mass flow and momentum rates of the inflowing wind, +as is discussed in Paper I. +The velocity components of the injected flow in the +2- and 3-directions are set so that the flow is along the +direction of the initial magnetic field lines. The injected +flow is also given an additional toroidal velocity compo- +nent: +vφ,inj = 0.23 +� rcyl +22.4r∗ +�−1/2 +vK∗. +(4) +The values employed for ri, ρ0, ˙m∗, ˙minj, and ˙pinj are +given for protostellar masses of 1, 2, 4, 8, 16, and 24 M⊙ +in Table 1. +In the smoothing region, at ri < r < ro, the veloc- +ity is gradually reduced by multiplying it by a factor +w = cos2[ π +2 (r − ri)/(ro − ri)]. The initial density of the +surrounding envelope is gradually joined with the den- +sity in the injection region by dividing the core density +by 1+w(fjump−1), where fjump is the ratio of the initial +core density to the density in the injection region. +We note that, especially in the outflow cavity, if the +density in a cell drops too low, the Alfv´en time step +drops to such a low value that the simulation effectively +grinds to a halt. To avoid this, it is common practice in +outflow simulations to implement a density floor, which +prevents the Alfv´en time step from becoming extremely +small. However, including such a density floor means +mass is being artificially added to the grid. In this work, +we have used a density floor that depends on height x1 +above the disk: nH,floor = (x1/105 au)−1 cm−3. The rea- +son for this choice is that near the inner x1 boundary +where mass is accreting, we need a fairly large density +floor to maintain a reasonable Alfv´en time step as the +magnetic fields are stronger. High above the disk, the +density in the outflow cavity drops to values much below +what the floor needs to be near the inner x1 boundary, +and hence the density floor in the outer part of the sim- +ulation box can be lower than in the inner part. We + +5 +Table 1. Values of the radius of the injection region ri, the injected density along the axis ρ0, the desired accretion rate ˙macc, +the desired injected mass flow rate ˙minj, and the desired injected momentum rate ˙pinj employed at the lower x1 boundary for +protostellar masses m∗. +m∗ +ri +ρ0 +˙macc +˙minj +˙pinj +[M⊙] +[au] +[10−17 g cm−3] +[10−4M⊙ yr−1] +[10−5M⊙ yr−1] +[10−3M⊙ km s−1 yr−1] +1 +92 +5.8 +1.0 +1.0 +5.9 +2 +106 +4.4 +1.4 +1.4 +9.9 +4 +124 +1.1 +2.0 +2.0 +9.4 +8 +150 +0.6 +2.7 +2.7 +14.2 +16 +196 +1.5 +3.2 +3.2 +41.2 +24 +282 +1.0 +3.3 +3.3 +49.5 +note that when mass is added to a cell in the simula- +tion, we do not adjust the velocity of that cell, and as a +consequence momentum is also added to the simulation. +3. RESULTS +3.1. Density, velocity and magnetic field structures +We have simulated the evolution of the protostellar +core for ∼ 105yr as the protostar grows from m∗ = 1M⊙ +to about 26 M⊙. In Fig. 1 we show slices of the density +structure in the x1 − x2 plane at x3 = 0. These images +show the general structure of the disk-wind outflow cav- +ity as it gradually carves open a larger and larger vol- +ume from the initial core infall envelope. +Concurrent +with this evolution of the outflow cavity, we also see the +collapse of the infall envelope down towards the central +midplane base of the core. A movie showing the evo- +lution of this structure is shown in Fig. 2. During the +course of the evolution the range of densities present in +the simulation extends from nH ∼ 4cm−3 (in the outflow +cavity) to ≳ 108 cm−3 (in the inner infall envelope). +Figure 3 shows the magnitude of the outflowing ve- +locity along the x1 direction, i.e., v1 > 0.9 km s−1, for +the same slices through the simulation domain shown +in Fig. 1. At any given evolutionary stage, the highest +velocities are found close to the central axis of the out- +flow cavity. At the earliest stages shown in Fig. 3, i.e., +m∗ = 2M⊙, these velocities are already ∼ 2, 000km s−1. +By the later stages with m∗ = 24 M⊙, these velocities +have risen to ∼ 5, 000 km s−1. +Figure 4 shows the magnitude of the total magnetic +field strength for the same slices through the simula- +tion domain shown in Fig. 1. The largest magnetic field +strengths are ∼ 100mG near the base of the outflow and +inner infall envelope. In the outflow cavity, the magnetic +field strength is much lower than in the infall envelope, +with values at low as ∼ 0.01 mG. +3.2. Evolution of the outflow cavity opening angle +To evaluate the opening angle of the outflow cavity +at a given height x1, we first calculate the area A in +the x2-x3 plane of the outflowing matter that has v1 > +0.9 km s−1. We then approximate the outflow as having +a conical shape with a circular cross section of area A = +πr2, giving r = +� +A/π, and then find the half opening +angle of that cone, tan(θoutflow) = r/x1. In Fig. 5 we +show the evolution of the calculated opening angle over +time for several different heights above the disk. These +direct estimates of the opening angles are stopped when +the outflow cavity region approaches the lateral edges +of the simulation domain. +Beyond this point, shown +with dashed lines, we make an approximate estimate for +opening angle at a given height via linear extrapolation +from the closest lower height where the geometry of the +outflow is still contained within the domain. +From our results we see that the outflow cavity open- +ing angle is larger at lower heights (e.g., at 5,000 au), +and is smaller at larger heights due to collimation of the +outflow. In other words, the outflow cavity is not truly +conical (as is evidenced in Figs. 1 and 2). Considering +a fidcuial height equal to the initial radius of the core, +i.e., 12,000 au, we see that the outflow cavity opening +angle has achieved a value of about 10◦ at the earliest +stages of the simulation, i.e., when m∗ = 2 M⊙. It then +rises slowly until m∗ ∼ 4 M⊙. After this it increases +at a slightly faster rate, reach about 42◦ by the time +m∗ = 18M⊙, i.e., the last stage where it can be directly +evaluated in the simulation domain. An extrapolation +based estimate at m∗ = 24 M⊙ yields θoutflow ≃ 50◦. +In Figure 5 we also compare our results to those of +Paper I (without pre-clearing), which were calculated +at the top of the grid in those simulations, i.e., at a +height of about 12, 000 au. Recall that in Paper I, with +models run at fixed m∗, it was somewhat uncertain at +which time to evaluate the results for the opening angle. +Paper I also considered a case “with pre-clearing” that +attempted to allow for the earlier stages of evolution and +these yielded larger opening angles at the later stages, +i.e., about 50◦ at m∗ = 16M⊙ and 78◦ at 24M⊙. We find +that our new simulations with a continuous evolution +followed from low to high values of m∗ yield moderately + +6 +Figure 1. Slices of simulation results for density in the x1 − x2 plane at x3 = 0, with x1 corresponding to the outflow axis. +The top, middle and bottom rows show m∗ = 2 M⊙ and 4 M⊙, 8 M⊙ and 12 M⊙, and 16 M⊙ and 24 M⊙, respectively. +smaller cavity opening angles than the results of Paper I, +with the biggest differences being at the highest masses. +We also compare our results to the opening angles +predicted by the semi-analytic model of Zhang et al. +(2014b), following the method of Matzner & McKee +(2000), which is based on the condition of whether the +material in a given direction can be accelerated to the es- +cape speed. We find that our numerical results predict a +moderately narrower outflow cavity geometry than this +semi-analytic model, with the difference being about 20◦ +by the end of the simulation. +3.3. Mass and momentum fluxes of the outflow +We evaluate the rate at which mass flows out of the +top of the simulation box at the x2 − x3 boundary face +via (1/2) ˙moutflow = +� +ρv1dA, i.e., performing the sum- +mation over the actual area of the outflow with no as- +sumption of it being circular and equating this to half +the total mass flux in a bipolar protostellar outflow. The +evolution of this outflowing mass flux is shown in Fig- +ure 6a. +Initially, there is a transient phase with a fairly high +mass flux out of the simulation box of ∼ 4×10−5M⊙yr−1 + +8.20 +2.5×104 +2×104 +[np] +1.5x104 +6.93 +x +104 +5000 +5.67 +2.5x104 +2x104 +[np] +4.40 +104 +5000 +3.13 +2.5x104 +t=94,000 yrs, M=24 +2x104 +[np] +1.5x104 +1.87 +x +104 +5000 +0.60 +-10000 +0 +10000 +-10000 +0 +10000 +Log(n/[cm-3]) +Lnp] +X2 +npl7 +Figure 2. Movie showing the temporal evolution of the x1 − x2 at x3 = 0 density slices, i.e., same as the examples shown in +Fig. 1. +while the outflow cavity is being cleared out. After this +the mass flow rate grows from about 2 × 10−5 M⊙ yr−1 +to ∼ 1 × 10−4 M⊙ yr−1 by the time the star has reached +∼ 10M⊙. We note that the mass flux exhibits moderate, +∼ 30%, fluctuations during this evolution. After this the +mass flux stops increasing and exhibits more dramatic +fluctuations during the evolution to m∗ = 16 M⊙. After +this, it shows a more steady, smooth decline, which is +mostly caused by the outflow cavity expanding beyond +the size of the top face of the simulation domain. For +this reason, we do not calculate the mass flow rate out +of the grid for masses beyond ∼ 20 M⊙: i.e., at this +stage a significant amount of mass is now leaving across +the side boundaries (as can be observed in the movie in +Fig. 2 and in Fig. 3). +Figure 6b shows the ratio of the mass flux leaving +the top of the simulation domain to the mass injected +at the base of the outflow. After the initial peak asso- +ciated with first breakout of the outflow, this ratio is +about 2, but then rises up to a peak just below 10 when +m∗ = 10M⊙. At higher masses it generally declines, but +with large fluctuations, eventually reaching values near +2 again. +Figure 6c shows the time evolution of the total mass +that has left the top of the simulation domain. We find +that more than 4 M⊙ has left the grid as part of the +outflow by the time the protostar reaches 20 M⊙. +Figure 7a shows the momentum flux passing through +the top of the simulation domain, evaluated as ˙p = +� +ρv2 +1dA. +As in Fig. 6, we cut off the measurements +when substantial mass and momentum start to leave +the domain through the side boundaries. We find that +the momentum flux leaving the domain stays approxi- +mately constant at about 0.005 M⊙ km s−1 yr−1, until +the star reaches ∼ 7 M⊙. +Then it increases to reach +nearly 0.02 M⊙ km s−1 yr−1 when the star is ∼ 16 M⊙. +It then continues to increase, but at a slower rate. How- +ever, at this stage we begin to lose track of mass that is +leaving through the sides of the domain. +Figure 7a also shows the injected momentum flux at +the base of the outflow. In general, as expected, we see +a very good agreement between the injected and ejected +momentum fluxes, with the largest deviation occurring + +8.20 +2.5x104 +6.93 +2x104 +5.67 +[αu] +1.5x104 +4.40 ++2 +104 +3.13 +5000 +1.87 +1.5x104-1x104-5000 +0 +5000 +104 +1.5×104 +0.60 +x, [αu] +28 +Figure 3. +Slices in the x1 − x2 plane at x3 = 0 of simulation results for total velocity, v, but only showing cells with +v1 > 0.9 km s−1 to highlight outflowing gas. The top, middle and bottom rows show m∗ = 2 M⊙ and 4 M⊙, 8 M⊙ and 12 M⊙, +and 16 M⊙ and 24 M⊙, respectively. +at late times due to some outflow material leaving via +the sides of the domain. The ratio of these momentum +fluxes is shown explicitly in Figure 7b. +Figure 7c shows the total momentum that has left via +the top of the simulation domain. This grows steadily +to reach ∼ 800M⊙ km s−1 by the time the protostar has +reached ∼ 20 M⊙. +3.4. Star formation efficiency +Here we evaluate the star formation efficiency (SFE), +i.e., the ratio of the final stellar mass to the initial core +mass, that is implied by our simulation results. After +100,000 years, the protostar has grown to m∗ ≃ 26 M⊙. +Thus we estimate that ¯ϵ∗f ≥ 0.43. +This is a lower +limit since in our model the disk has a mass of mdisk = +(1/3)m∗ ≃ 9 M⊙ and a significant portion of this ma- +terial is expected to be able to accrete to the star. If +the only process diverting material from the accretion +disk is injection into the disk wind with ˙mw = 0.1 ˙m∗, +then the final stellar mass would be at least 34 M⊙, i.e., +¯ϵ∗f ≥ 0.56. It is possible that a larger fraction of ma- +terial could be diverted from the accretion disk if other +forms of feedback, especially disk photoevaporation, are +significant. +However, Tanaka et al. (2017) considered +such models and found that disk photoevaporation was +relatively unimportant compared to the disk wind mass +flux for this mass and accretion rate regime. +The above estimates are likely to still be lower limits, +since there is still 12M⊙ (3M⊙ from the initial core and +9 M⊙ from the surrounding clump) remaining in the + +2.5×10 +t=9,000 yrs. +t=21,000 +3.70 +[np] +yrs. +M=2 +M= 4 +2.95 +5000 +t=39,000 +t=54,000 +2x104 +2.20 +[no +1.5x104 +yrs, +yrs, +M=8 +M=12 +5000 +W +1.45 +2.5x104 +t=68,000 yrs. +t=94,000 +2x104 +1.5x104 +yrs, +0.70 +104 +M=16 +M=24 +5000 +@ +10000 +0.05 +-10000 +0 +-10000 +0 +10000 +X2 +[np] +X2 +[au] +log(v/[km s-1])9 +Figure 4. Slices of simulation results for magnetic field strength, B, in the x1 − x2 plane at x3 = 0. The top, middle and +bottom rows show m∗ = 2 M⊙ and 4 M⊙, 8 M⊙ and 12 M⊙, and 16 M⊙ and 24 M⊙, respectively. +simulation domain, i.e., 24 M⊙ in the global, mirrored +domain. One expects that a significant fraction of this +material would be accreted to the central protostar. In +the case that all of the remaining initial core mass is +accreted, i.e., 6 M⊙, then this would thus result in a +SFE of ¯ϵ∗f ≃ 0.67. +Comparing the semi-analytic model of Zhang et al. +(2014b), they also reached a final value of m∗ = 26 M⊙. +Thus, with the same considerations of residual disk ac- +cretion, they expect to reach ¯ϵ∗f ≥ 0.56. However, their +model at this point would be exhausted of gas and so +this would be the final estimate of SFE. Thus we con- +clude that the expected SFE from our numerical model +is moderately (∼ 20%) larger than that predicted by +the semi-analytic model. +This is consistent with the +generally smaller outflow opening angles found during +the course of the evolution in the numerical model com- +pared the Zhang et al. (2014b) semi-analytic model (see +Fig. 5). +However, we note that in the fiducial TCA model of +McKee & Tan (2003), the initial core is expected to in- +teract with significant surrounding clump gas during its +collapse to a protostar, so with this consideration the +results of Zhang et al. (2014b) for the final stellar mass, +m∗f, are also lower limits. If SFE is defined with respect +to the initial core mass, then the values of ¯ϵ∗f would also +be lower limits. + +2.5x104 +一 +1.00 +2x104 +[np] +1.5x104 +104 +5000 +一 +-2.00 +2.5x10* +2×104 +[np] +1.5×104 +-3.00 +104 +5000 +2.5x104 +4.00 +[np] +1.5x104 +x +104 +5000 +-5.00 +-10000 +0 +10000 +-10000 +0 +10000 +X2 +[np] +×2 [αu] +Log(B/[G])10 + 0 + 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 0 + 5 + 10 + 15 + 20 + 25 +θoutflow [degrees] +m* [M⊙] +Staff et al. (2019) +Zhang et al. (2014) +height 5,000 au +height 12,000 au +height 20,000 au +height 25,000 au +12,000 au extrapolated +20,000 au extrapolated +25,000 au extrapolated +Figure 5. Outflow cavity opening angle measured at different heights above the disk (solid lines). Extrapolated estimates +(dashed lines) are needed once the cavity nears the simulation boundary at a given height (see text). Also shown are the outflow +cavity opening angles found in the numerical models of Paper I (squares) and the semi-analytic models of Zhang et al. (2014b) +(crosses). +3.5. Outflow mass spectra +One method of comparing our model results with ob- +served systems is via the distribution of outflowing gas +mass with line of sight velocity velocity, i.e., “mass spec- +tra”, since this can be inferred from observations of CO +emission lines. +Note, in this paper we will not make +synthetic CO spectra of our models, deferring this step +to a future work. To produce the distribution of mass +with line of sight velocity, we need to produce a “global” +simulation domain, which is achieved by mirroring our +simulation grid about the x1 = 0 boundary, i.e., the +disk plane. In this way we produce a symmetric bipolar +outflow structure, which we then view at various angles, +θview, to the outflow axis. Note, θview = 0◦ is defined as +a line of sight that is parallel to the outflow axis. +Figure 8 shows the mass spectra within the global do- +main at various evolutionary stages. Note, these spectra +include all gas, i.e., both outflowing and infalling mate- +rial. We have chosen three values of θview that are part +of the grid of uniformly sampled grid of cos θview values +in the radiative transfer models of Zhang & Tan (2018). +The mass spectra show a sharp peak at low velocities, +and, except for θview values close to 90◦, long tails to +larger velocities. As the protostellar mass increases, we +find more mass at larger velocities. For m∗ > 16 M⊙, +the largest velocities are > 3000km s−1 when the system +is viewed close to the outflow axis. One point to note is +that between 2 M⊙ and 4 M⊙, the maximum velocities +decrease somewhat. This is due to the protostellar ra- +dius (which also sets the inner disk radius) growing from +3.45 R⊙ at 2 M⊙ to 20.5 R⊙ at 4 M⊙. The injection ve- +locity of the outflow is proportional to the Keplerian +speed at the launching point (vKep ∝ m1/2 +∗ +r−1/2; Eq. +3). Hence, the highest velocity outflow is launched from +the inner disk and, as the inner disk radius expands, the +velocity of the material launched from the inner disc de- +creases, even though the central mass is growing. We +use these mass spectra in the next subsection to make +detailed comparisons to some observed massive proto- +stars. +3.6. Comparison with observed outflow mass spectra +In Figures 9 and 10 we compare the simulation out- +flow mass spectra to equivalent outflow mass spectra +of G35.20-0.74N and G339.88-1.26 (hereafter G35.2 and +G339) as derived from ALMA observations of CO(2- +1) line emission by Zhang et al. (2022) and Zhang +et al. (2019), respectively. Note, the observed line emis- +sion from these sources was extracted from regions of + +11 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 + 0 + 5 + 10 + 15 + 20 +1/2 m +. + outflow [10-4 M☉ yr-1] +m* [M⊙] + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 0 + 5 + 10 + 15 + 20 +m +. + outflow/m +. + inj +m* [M⊙] + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 + 4.5 + 0 + 5 + 10 + 15 + 20 +∫ 1/2 m +. + outflow dt [M☉] +m* [M⊙] +Figure 6. (a) Top: Evolution of outflow mass flux through +the top of the simulation domain (x2 − x3 face at x1 = +25, 000 au) (purple solid line). +The red dashed line shows +the injected mass flow rate of the outflow. (b) Middle: Ratio +of the mass flow rate out of the top of the simulation box to +the injected mass flow rate at base of the outflow. (c) Top: +Evolution of total mass that has left the top of the simulation +domain by being swept-up by the outflow. +∼25,000 au in radial size centered on the protostars, +similar to the size of our simulation box. We consider +a velocity range of ±50 km s−1 and exclude the inner +±10 km s−1, which is affected by the presence of ambi- +ent clump gas. +To quantify the differences between the models and +observations, we calculate the reduced χ2 between the +two, following the method of Zhang & Tan (2018) (de- +veloped for spectral energy distribution fitting), as: +χ2 = 1 +N +� +i +�mi,data − mi,sim +σ +�2 +, +(5) + 0 + 0.005 + 0.01 + 0.015 + 0.02 + 0.025 + 0 + 5 + 10 + 15 + 20 +1/2 p +. + outflow [M☉ km s-1 yr-1] +m* [M⊙] + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 + 1.6 + 1.8 + 0 + 5 + 10 + 15 + 20 +p +. + outflow/p +. + inj +m* [M⊙] + 0 + 100 + 200 + 300 + 400 + 500 + 600 + 700 + 800 + 900 + 0 + 5 + 10 + 15 + 20 +∫ 1/2 p +. + outflow dt [M☉ km s-1] +m* [M⊙] +Figure 7. (a) Top: Evolution of outflow momentum flux +through the top of the simulation domain (x2 − x3 face at +x1 = 25, 000 au) (purple solid line). +The red dashed line +shows the injected momentum flux at the base of the outflow. +The green solid line shows the momentum flux injected in +the semi-analytic model of Zhang et al. (2014b). (b) Middle: +Evolution of the ratio of the momentum flux through the top +of the simulation domain to the injected momentum flux at +the base of the outflow. (c) Bottom: Evolution of the total +momentum that has left the top of the simulation domain. +where N is the number of data points, mi,data and mi,sim +are the mass in the i’th velocity bin in the observed data +and in the simulation, and σ is the uncertainty on the +observed data. The uncertainty in the data is assumed +to be comprised of a systematic uncertainty of 40% and +a noise level that is ∼ 6 × 10−5 M⊙/(km s−1) (for both +G35.2 and G339). Note that while the mass spectra are +shown in log space, we perform the χ2 fitting in linear +space. + +12 +Figure 8. Distribution of outflow mass with line of sight velocity for material within a global (i.e., mirrored) simulation domain +at various evolutionary stages (i.e., protostellar masses) and as viewed at different inclination angles, θview = 12.8◦, 61.4◦, 88.6◦. +As seen in Figure 9, G35.2’s outflow mass spectrum at +negative velocities is affected by a significant absorption +feature at −20km s−1, which may be due to other molec- +ular cloud components along the line of sight. Thus, for +this source we restrict fitting to only the positive veloc- +ity range. Figure 10 shows that G339’s mass spectrum +at positive velocities is similarly affected by absorption +features and so here we only fit to the negative velocity +range. +Each of the panels in Figures 9 and 10 shows the mod- +els at a particular evolutionary stage as seen over the full +range of viewing angles, i.e., uniformly sampling cosθview +from 0.025 to 0.975 in steps of 0.05. We can see that at +small values of m∗ the models generally fail to to match +the observational data. In particular, they underpredict +the amount of outflowing gas at low and intermediate +velocities. For G35.2, there is a better agreement in the +shape of the mass spectrum when m∗ ∼ 16M⊙ to 24M⊙, +although the model is systematically low by a factor of +about 3. For G339, the shape of the mass spectrum has +a best match when m∗ ∼ 20 M⊙, but is again low be +about a factor of 3. We note that such systematic off- +sets could be explained, at least in part, by uncertainties +in the conversion of CO(2-1) line flux to mass. The dif- +ference could also simply be due to the observed systems +being more massive protostellar cores, i.e., involving an +initial core mass that is > 60 M⊙. Within the context +of the Turbulent Core Accretion model, there is also the +additional parameter of Σcl, which could be varied from +the fiducial value of 1 g cm−2 assumed here. +Given the above considerations, we do not attempt +to adjust our models further to find a better match to +the data, since such a step will likely require running a +much larger grid of simulations to explore the Mc and +Σcl parameter space. Nevertheless, with the context of +the models we have presented, there is formally a best +fitting model for each of G35.2 and G339. To illustrate +these and the dependence of χ2 on model parameters, in +Figure 11 we plot χ2 versus cos θview for all the consid- +ered models at various evolutionary stages. Again, we +can see that the observations are more consistent with +higher protostellar masses. +However, in these higher +mass cases, we note that the goodness of fit does not +depend very sensitively on the viewing angle. + +c0s(0)=0.025 (0=88.69 +cos(8)=0.475 (8=61.4)) +cos(9)-0.975 (0-12.80) +2 +M=2 M +M=1E M +4 +_ ) +[ +2 +4 +M=8 +8 +2000 +0 +2000 +2000 +0 +2000 +v[km s +v[krm s-]]13 +Figure 9. The mass velocity spectra from the simulation compared to that from observations of G35.20-0.74N (Zhang et al. +2022) for velocities less than ±50 km s−1. +3.7. Comparison to other observational metrics of +massive protostars +The mass flow rate out of the simulation box (see +Fig. 6) starts out at a few ×10−5 M⊙ yr−1 for the first +∼ 50, 000 years until the star reaches ∼ 10 M⊙, before +increasing to more than 10−4 M⊙ yr−1 and becoming +quite variable during the latter parts of the simulation. +The momentum flux out of the simulation box (Fig. 7) +is, meanwhile, about 5 × 10−3 M⊙ km s−1 yr−1 for the +first ∼ 40, 000 years until the star reaches ∼ 8 M⊙, +after which the momentum rate grows steadily to ∼ +2 × 10−2 M⊙ km s−1 yr−1, and also shows time-variable +behaviour. Such values are in general agreement with +observations of outflows from massive protostars (Wu +et al. 2004; Maud et al. 2015; Fedriani et al. 2019), al- +though it should be noted that there are significant un- +certainties associated with the observational derivation +of these mass and momentum fluxes. +There have been a few measurements of magnetic field +strengths in the outflows of massive protostars. In Orion +Source I, which is thought to be 10 − 20 M⊙ protostar +(e.g., see discussion in Hirota et al. 2020), the magnetic +field strength was estimated to be 30 mG on a scale of +a few hundred au. This is in reasonable agreement with +our simulations on similar scales (Fig. 4). +4. DISCUSSION +4.1. Comparison with previous simulation studies +Here we discuss how our simulation results to those of +other relevant studies of massive star formation, mostly +restricting our consideration to those including pro- +tostellar outflow feedback with magnetohydrodynamic +(MHD) simulations. The simulation we have presented, +in addition to its initial core, has a well defined boundary +condition during the evolution for the input protostel- +lar outflow, which is tied to the evolution of the fidu- + +M=2 +M +M=16 +M +2 +4 +1 +M=20 +2 +4 +M=24 +2 +4 +M=12 +Mo +cos(0) +了 +4 +5 +0.00 +0.17 +0.33 +0.50 +0.67 +0.83 +1.00 +-40 +-20 +0 +20 +40 +Ikm s14 +Figure 10. The mass velocity spectrum from the simulation compared to that from observations of G339.88-1.25 (Zhang et al. +2019), for velocities less than ±50 km s−1. +cial massive protostar in the Turbulent Core Accretion +model (McKee & Tan 2003; Zhang et al. 2014b). One +comparable non-MHD simulation is that of Kuiper & +Hosokawa (2018), who presented a simulation of a mas- +sive protostar forming from a surrounding mass reservoir +from 100 M⊙ to 1000 M⊙. The simulation code Pluto +was utilized with a logarithmically spaced spherical co- +ordinate grid assuming axial and midplane symmetry +of the system. Feedback from radiation pressure, ion- +ization and injected protostellar outflows was included. +However, the simulation did not include magnetic fields. +In contrast, the following simulation studies generally +present collapse of a fully 3D gas structure to a sink +particle representing a protostellar source. For example, +Rosen & Krumholz (2020) performed radiation MHD +simulations of a collapsing 150 M⊙ core (significantly +more massive than the 60 M⊙ core we consider in this +study), and followed the evolution until the star reached +a mass of 33.64 M⊙. They found that once the stellar +mass reached about 30 M⊙, radiation pressure created +by the central star starts driving an expanding bubble. +Radiative effects like this could potentially be relevant +in our case if we continued the simulation beyond 30M⊙ +(see also Tanaka et al. 2017). +Commer¸con et al. (2021) compared collapse simula- +tions of a 100 M⊙ core in several scenarios: without +magnetic fields, with ideal MHD, and with ambipolar +diffusion. In the case of the non-magnetized simulation, +they found a very weak outflow dominated by episodes +of accretion bursts. In their ideal MHD simulation, they +found that an increased pressure in the central region, +due to increased stellar luminosity and build-up of mag- +netic field, causes the outflow to almost disappear when +the protostar reaches ∼ 10M⊙. However, this behaviour +is not observed in their non-ideal MHD simulation. + +1 +M=2 +M= 16 M +2 +4 +1 +M=4 +M +M=20 M ++ +4 +1 +M=8 +M +M=24 +3 +4 +1 +2 +cos(0) +4 +L +0.00 +0.17 +0.33 +0.50 +0.67 +0.83 +1.00 +-40 +-20 +0 +20 +40 +[km s15 +Figure 11. Dependence of χ2 derived from fitting our sim- +ulated mass spectra for different evolutionary stages (i.e., +various values of m∗) to the observational data of massive +protostars G35.2 (top) and G339 (bottom) as a function of +the cosine of the viewing angle. +Mignon-Risse et al. (2021b,a) performed radiation +MHD collapse simulations also of a 100 M⊙ core. +Mignon-Risse +et +al. +(2021a) +focused +on +the +out- +flow. +They found mass outflow rates of ∼ 10−5 − +10−4 M⊙ yr−1. The momentum rate that they found +was ∼ 10−4M⊙km s−1 yr−1, which is much smaller than +the ∼ 10−3 − 10−2 M⊙ km s−1 yr−1 that we measure in +our simulation. We also note that our model involves +the momentum rate growing as the protostellar mass +grows, while they found a roughly constant momentum +rate with time. Also, contrary to our work, the opening +angle in their simulations for the most part decreased +with time. +4.2. The role of the magnetic field +In ideal MHD, the gas is forced to follow the field lines. +This therefore creates a natural separation between the +outflowing gas and the collapsing envelope, because the +field lines found in the outflow are anchored in the in- +jection region. To demonstrate this we performed a test +simulation with the same set up, but without magnetic +field. In Fig. 12, we show slices of the density structures +and velocity fields of the outflowing gas for simulations +with and without magnetic field after 39,000 years (i.e., +when the protostar has reached 8 M⊙). A consequence +of the lack of magnetic field is less collimated, slower +outflow, which interacts with much more envelope ma- +terial, causing a larger mass flow rate out of the simu- +lation box as more envelope material is entrained in the +outflow. We also find that the outflow cavity is much +less distinct, i.e., in its density contrast with the infall +envelope, in the simulation without magnetic field. Be- +cause of this, there is no high-velocity outflow, and the +momentum flow rate at a height of 25,000 au is smaller +than in the simulation with magnetic field. Interestingly, +the outflow pushes more material sideways when there is +no magnetic field to confine it, forcing envelope material +farther away from the protostar where the gravitational +force is weaker, causing the envelope to collapse more +slowly. As a consequence, the envelope “puffs up” side- +ways in the no-magnetic field simulation, and at 39,000 +years it extends beyond the side boundaries (see density +panels in Fig. 12). +4.3. Effect of numerical resolution +To examine the dependence on numerical resolution, +we ran the same simulation set up with twice as many +cells in each direction (i.e., 336×560×560 cells; see §2.1), +but keeping other parameters the same. In this higher +resolution simulation, the smallest cells are now roughly +6 au on each side, compared to roughly 12 au in our +primary “medium” resolution simulation. This higher +resolution simulation is much more computationally ex- +pensive, and it was not feasible to run it for the entire +evolution (i.e., up to ∼ 24M⊙). Instead, we compare the +results between the two resolutions at t = 39, 000 years, +when the star has reached 8M⊙. In Fig. 13, we compare +the logarithm of the number density, and the velocity +field of the outflowing gas (where v1 > 0.9 km s−1), in +a slice through the middle of the grid (x3 = 0). +The medium and high resolution simulations are qual- +itatively and quantitatively similar. For example, the +opening angle of the outflow in the high resolution simu- +lation measured at 12,000 au is 17.0◦, compared to 20.0◦ +in the medium resolution simulation. Note, while the +low density part of the outflow cavity appears slightly +larger in the slice of the high resolution simulation shown +in Fig. 13, the cavity defined by the outflowing gas is in +fact slightly smaller. At 39,000 years, in the high resolu- +tion simulation we find that 1.5 M⊙ has left the simula- +tion box with the outflow through the outer x1 bound- +ary, while in the medium resolution simulation 1.2 M⊙ +has left the box. These example diagnostics indicates a +fairly good agreement between the higher and medium +resolution simulations. + +6 +M: +M +M= +5 +M= +M: +20 +24 +4 +2. +3 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +cos(0)6 +M: +248 +M: +M= +M= +5 +M= +M: +20 +24 +4 +2. +3 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +cos(0)16 +Figure 12. The effect of magnetic fields on the outflow structure is illustrated by a comparison of the number density in the +x1 − x2 slice at x3 = 0 and time 39,000 years, when the protostar is 8 M⊙ for a case without magnetic field (|B| = 0) (left +panels) and with a magnetic field (i.e., our fiduical model) (right panels). The upper panels show density structure; the lower +panels show the velocity field of the outflowing gas. + +Log(n/[cm-3]) +0.60 +1.87 +3.13 +4.40 +5.67 +6.93 +8.20 +time=39,000 yeurs +Without B-field +Medium resolution +2.5x104 +2x104 +[np] +1.5x104 +x +104 +5000 +-10000 +0 +10000 +-10000 +0 +10000 +X +[au] +X +Inplog(v/[km s-1]) +-0.05 +0.63 +1.31 +1.99 +2.67 +3.35 +time=39,000 ye0rs +Without B-field +with B-field +2.5x10 +2x104 +x +104 +5000 +10000 +0 +10000 +-10000 +0 +10000 +X> +[nD17 +Figure 13. Effect of numerical resolution is illustrated by a comparison of the density structure in the x1 − x2 plane at x3 = 0 +at 39,000 years (m∗ = 8 M⊙) for the high resolution simulation (left panels) and fiducial medium resolution simulation (right +panels). The upper panels show density structure; the lower panels show the velocity field of the outflowing gas. +5. CONCLUSIONS +We have presented a 3D-MHD simulation of a +magnetically-powered disk wind outflow from a massive +protostar located at the center of a core with initial mass +of 60 M⊙ and radius of 12,000 au. Such a core is the +fiducial case of the Turbulent Core Accretion model of +McKee & Tan (2003), which involves the core being pres- +sure confined by an ambient clump medium with mass +surface density of Σcl = 1 g cm−2. We have followed the +evolution for 100,000 years as the protostar grows from +m∗ = 1 M⊙ to about 26 M⊙, following the protostellar +evolutionary track of Zhang et al. (2014b), which sets +both the accretion rate to the star and the mass and +momentum injection rate to the disk wind outflow. +We find that the protostar drives a powerful, colli- +mated outflow that breaks out of the core at relatively +early times, i.e., within ∼ 1, 000 yr of the start of the +simulation. At the scale of the initial core, the outflow +has an opening angle (from outflow axis to cavity edge) +of just over 10◦ until m∗ = 4 M⊙ at 21,000 yr. There- +after, as the protostar grows in mass and contracts to- + +Log(n/[cm-3]) +0.60 +1.87 +3.13 +4.40 +5.67 +6.93 +8.20 +time=39,000 yeurs +High resolution +Medium resolution +2.5x104 +2×104 +[nD +1.5x104 +x +104 +5000 +-10000 +0 +10000 +-10000 +0 +10000 +Lau] +X2 +Inplog(v/[km s-1]) +-0.05 +0.63 +1.31 +1.99 +2.67 +3.35 +time=39,000 yeurs +High resolution +Medium resolution +2.5x10 +2x104 +x +104 +5000 +10000 +0 +10000 +-10000 +0 +10000 +Lau] +X> +au18 +wards the zero age main sequence, the outflow becomes +more powerful causing the cavity to open up gradually, +reaching opening angles of about 50◦ by the end of the +simulation. This disk wind outflow feedback thus dra- +matically affects the density structure and morphology +of the protostar. While we have not performed radia- +tive transfer (RT) calculations on these simulations (de- +ferring this step for a future work), the RT models of +Zhang et al. (2014b) based on a semi-analytic core and +outflow structure already illustrate the importance of +such cavities for determining the infrared images and +SEDs of the protostars. +The outflow also is the main factor determining the +star formation efficiency (SFE) from the core. We find +a lower limit to this SFE of ¯ϵ∗f = 0.43, but, considering +the presence of a massive accretion disk and residual +infall envelope, we estimate that the final value could +reach as high as ¯ϵ∗f ≃ 0.7. Such values are moderately +higher than the efficiencies assumed of 0.5 in the fiducial +TCA model of McKee & Tan (2003). +Inside the outflow cavity we find that the magnetic +field is relatively weak, ∼ 10−4−10−5 G, while it retains +its initial core value ∼ 10−3 G just outside the outflow +cavity. Near the base of the outflow, however, we find +magnetic field strengths of ∼ 0.1 G. The magnetic field +structure we have implemented acts to help separate the +outflow from the collapsing core, limiting the amount of +the envelope material being entrained in the outflow. +The mass flow and momentum rates of our simu- +lation are ∼ 2 × 10−5 − 2 × 10−4 M⊙ yr−1 and ∼ +2 × 10−3 − 2 × 10−2 M⊙ km s−1 yr−1 respectively, with +these values controlled by the boundary conditions we +have implemented, but also comparable to rates mea- +sured from observed massive protostars. We have also +compared the distribution of outflow mass with veloc- +ity, i.e., outflow mass spectra, of our simulations out +to velocities of ±50 km s−1 with two example massive +protostars G35.2 and G339 observed by ALMA. This +comparison indicates that such observations have di- +agnostic power to constrain model parameters related +to evolutionary stage, i.e., m∗, and viewing angle, i.e., +θview. While precise agreement between model and ob- +servation is not found (and is not expected given po- +tential systematic uncertainties in measure mass from +CO line emission and from the limited range of TCA +model parameters explored in our simulation), we do +find quite striking agreement in the shape of the out- +flow mass spectra for some models. Further diagnostic +tests involving full synthetic position-position-velocity +cubes of synthetic CO line emission will be presented in +a follow-up paper. +JES, JPR and JCT acknowledge support from Collab- +orative NSF grant AST-1910675. JES also acknowledges +support from NASA through grant HST-AR-15053 from +the Space Telescope Science Institute, which is operated +by AURA, Inc., under NASA contract NAS 5-26555. +JPR also acknowledges support from the Virginia Ini- +tiative on Cosmic Origins (VICO). JCT also acknowl- +edges support from ERC Advanced Grant MSTAR. +We acknowledge the use of NASA High-End Comput- +ing (HEC) resources through the NASA Advanced Su- +percomputing (NAS) division at Ames Research Cen- +ter to support this work. +The analysis and the fig- +ures have been made using GDL (Coulais et al. 2010), +VisIt: +https://visit-dav.github.io/visit-website/ , and +Gnuplot: http://www.gnuplot.info/ . +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +REFERENCES +Anderson, J. M., Li, Z.-Y., Krasnopolsky, R., & Blandford, +R. D. 2006, ApJL, 653, L33, doi: 10.1086/510307 +Arce, H. G., Shepherd, D., Gueth, F., et al. 2007, +Protostars and Planets V, 245 +Bally, J. 2016, ARA&A, 54, 491, +doi: 10.1146/annurev-astro-081915-023341 +Beltr´an, M. T., & de Wit, W. J. 2016, A&A Rv, 24, 6, +doi: 10.1007/s00159-015-0089-z +Beuther, H., Schilke, P., Sridharan, T. K., et al. 2002, +A&A, 383, 892, doi: 10.1051/0004-6361:20011808 +Beuther, H., Soler, J. 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C., et al. 2022, ApJ, +936, 68, doi: 10.3847/1538-4357/ac847f + diff --git a/5NAyT4oBgHgl3EQf2PnW/content/tmp_files/load_file.txt b/5NAyT4oBgHgl3EQf2PnW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..38ac84bd0018417a2c23afa547b3cf28280d0014 --- /dev/null +++ b/5NAyT4oBgHgl3EQf2PnW/content/tmp_files/load_file.txt @@ -0,0 +1,1315 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf,len=1314 +page_content='Draft version January 3, 2023 Typeset using LATEX preprint style in AASTeX631 Disk Wind Feedback from High-mass Protostars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The Evolutionary Sequence Jan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Staff,1 Kei E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Tanaka,2 Jon P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Ramsey,3 Yichen Zhang,3 and Jonathan C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Tan4 1Department of Space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Earth & Environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Chalmers University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Gothenburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Sweden and College of Science and Math,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' University of the Virgin Islands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' St Thomas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 00802,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' United States Virgin Islands 2Center for Astrophysics and Space Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Department of Astrophysical and Planetary Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' CO 80309,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' USA and ALMA Project,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' National Astronomical Observatory of Japan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Mitaka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Tokyo 181-8588,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Japan 3Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' University of Virginia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Charlottesville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' VA 22904,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' USA 4Department of Space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Earth & Environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Chalmers University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Gothenburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Sweden and Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' University of Virginia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Charlottesville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' VA 22904,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' USA (Dated:) ABSTRACT Star formation is ubiquitously associated with the ejection of accretion-powered outflows that carve bipolar cavities through the infalling envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' This feedback is expected to be important for regulating the efficiency of star formation from a natal pre-stellar core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' These low-extinction outflow cavities greatly affect the appearance of a protostar by allowing the escape of shorter wavelength photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Doppler-shifted CO line emission from outflows is also often the most prominent manifestation of deeply embedded early-stage star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Here, we present 3D magneto-hydrodynamic simulations of a disk wind outflow from a protostar forming from an initially 60 M⊙ core embedded in a high pressure environment typical of massive star-forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We simulate the growth of the protostar from m∗ = 1 M⊙ to 26 M⊙ over a period of ∼100,000 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The outflow quickly excavates a cavity with half opening angle of ∼ 10◦ through the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' This angle remains relatively constant until the star reaches 4 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' It then grows steadily in time, reaching a value of ∼ 50◦ by the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We estimate a lower limit to the star formation efficiency (SFE) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' However, accounting for continued accretion from a massive disk and residual infall envelope, we estimate that the final SFE may be as high as ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We examine observable properties of the outflow, especially the evolution of the cavity opening angle, total mass and momentum flux, and velocity distributions of the outflowing gas, and compare with the massive protostars G35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='20-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='74N and G339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='88-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='26 observed by ALMA, yielding constraints on their intrinsic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' INTRODUCTION Low-mass stars and their associated accretion disks form from gravitationally bound cores (Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 1987) and are frequently associated with the launching of bipo- lar jets and outflows (for reviews, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Bally 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The magnetocentrifugal mechanism (Blandford & Payne 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Pudritz & Norman 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Konigl & Pudritz 2000) is widely thought to be respon- Corresponding author: Jan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Staff jestaff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='astro@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='com sible for launching, accelerating and collimating these protostellar outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In this scenario, the combination of large-scale magnetic fields with gravity and rotation results in the ejection, acceleration and then collima- tion of gas originating from the surface of the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' A number of numerical simulation studies have been performed to investigate this process across a va- riety of different conditions and assumptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', Shi- bata & Uchida 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Uchida & Shibata 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Ouyed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2003, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Ouyed & Pudritz 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Romanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Zanni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Te¸sileanu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Sheikhnezami arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='00749v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='GA] 2 Jan 2023 2 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Stute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Stepanovs & Fendt 2014, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Ramsey & Clarke 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Gressel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Mat- tia & Fendt 2020a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' However, alternative theoreti- cal scenarios have also been proposed as being relevant for outflow launching, including: the X-wind model in- volving the interaction of the protostellar magnetic field with the inner disk (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', Lovelace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' stellar wind driven outflows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', Matt & Pudritz 2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' and magnetic pressure driven outflows (Lynden- Bell 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Observationally, support for the disk wind model in low- and intermediate-mass systems has been provided by high angular resolution observations of a handful of systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', TMC1A (Bjerkeli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2016), HH212 (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2017), DG Tau B (de Valon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2020), and IRAS 21078+5211 (Moscadelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In each case, the launching of the outflow can be traced to the accretion disk, demonstrating a launching radius that extends out to scales of up to ∼ 20 au from the central protostar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The formation of high-mass stars is more difficult to characterize observationally as there are fewer sources, they are farther away and they are more obscured by surrounding gas and dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Nevertheless, massive star formation is also typically observed to be associated with the launching of bipolar jets and outflows (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', Arce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Beltr´an & de Wit 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Hirota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' For example, the central source powering HH 80 and HH 81 (IRAS 18162-2048) (Marti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 1993), and G339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='88-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='26 (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2019) are both associated with highly collimated outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' An- other massive protostar, G35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='20-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='74N, has also been found to launch a highly collimated jet (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', Fedriani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Indeed, Caratti o Garatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2015) found that outflows from a number of intermediate and high- mass protostars appear as scaled-up versions of those from low-mass protostars, while Sandell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2020) also found this to be the case for the outflow from the massive protostar NGC 7538 IRS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Wider angle molec- ular outflows have also been observed from massive pro- tostars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Beuther et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2013, 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Maud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In general, the trend is that higher luminosity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', more massive, pro- tostars tend to have more powerful and more massive outflows with wider opening angles than their low-mass counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' McKee & Tan (2002) suggested that a combination of turbulence and magnetic pressure provides most of the support in a massive pre-stellar core against gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In this “Turbulent Core Accretion” (TCA) model, high- mass star formation is a scaled-up version of low-mass star formation, with accretion rates expected to be ∼ 10−4 to ∼ 10−3 M⊙ yr−1, compared to ∼ 10−6 to ∼ 10−5 M⊙ yr−1 in lower-mass cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' If that is the case, then outflows from forming massive stars can therefore also be a scaled-up version of the outflows from lower- mass forming stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Other formation scenarios for high-mass stars have also been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Bonnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (1998) suggested that high-mass stars form by the collision of multiple smaller objects that formed close together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Another possibil- ity suggested by Bonnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2001) is that massive stars form together in the central region of dense proto- clusters, where most of the mass is accreted from a glob- ally collapsing clump (see Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2014 for a review of these scenarios).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' This could lead high-mass stars to accrete from smaller disks that change orientation over time, leading to outflows that also keep changing direc- tions (Goddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In contrast to outflows from low-mass protostars, it is still debated whether or not strong magnetization is required to drive an outflow from high-mass protostars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' For example, Machida & Hosokawa (2020) found that, in their simulations, the outflow launching failed or was much delayed unless the initial cloud was strongly mag- netized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In contrast, Beuther et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2020), based on ob- servations, argued for a weak magnetization in the case of G327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='3, despite it also having an outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The direc- tion of the magnetic field in the core is also debated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' in some cases, it has been found to be parallel to the out- flow and perpendicular to the disk (Carrasco-Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Sanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2015), while other studies have found that the outflow axis is randomly oriented with respect to the core-field (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2014a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' From an analysis of about 200 outflows, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2022) find evidence for preferential alignment of outflow directions with large-scale B−fields, but with significant scatter for any given outflow to B−field to orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Staff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2019) (hereafter Paper I) presented 3D magneto-hydrodynamic (MHD) simulations of disk wind outflows from a 60 M⊙ core, but with the protostellar mass, accretion rate and mass outflow rate held at fixed values representing various stages of the protostellar evo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The method was to run each simulation for a roughly a local accretion time to infer the properties of the outflow cavity - envelope system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' However, because of this approximation this method involved significant uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In this paper, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', Paper II, we present similar MHD simulations as Paper I, but now following the protostel- lar evolutionary sequence consistently, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', as its mass grows from m∗ = 1 M⊙ to more than 24 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' As in Paper I, we assume that a star is growing from a 60 M⊙ core embedded in a clump with mass surface density of Σcl = 1 g cm−2 within the framework of the Turbulent 3 Core Accretion model of McKee & Tan (2002, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' A disk-wind (launched from the accretion disk) is injected into the simulation box, where some envelope material becomes entrained by the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We simulate the out- flow as it propagates through the envelope to investigate the interaction between the wind and the envelope ma- terial, and to investigate how much envelope material is pushed away, providing us with an estimate of the star formation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We also compare our simula- tion results with observations of outflows from massive protostars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In §2 we describe our numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We present our results in §3 and discuss their implications in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Finally, we summarize our findings in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' METHODS The goal of this work is to simulate a magnetically- powered outflow from a massive, growing protostar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Us- ing the ZEUS-MP code (Norman 2000), we conduct a 3D, ideal MHD simulation of an outflow from a massive protostar in the framework of the turbulent core accre- tion model (McKee & Tan 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2014b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Zhang & Tan 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' As in Paper I, we consider an initial core of mass of 60 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' However, instead of sim- ulating a sequence of separate models for different fixed values of protostellar mass, m∗, here we follow the evo- lution of a single simulation and the resulting outflow for more than 100,000 years as the central star grows from an small initial mass of m∗ = 1 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The setup of the simulation is described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Simulation domain and boundary conditions We use a Cartesian grid with 168 × 280 × 280 cells in the x1, x2, and x3 directions, respectively, for our “medium” resolution simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' A “high” resolution simulation is also run for the earlier phases of the evolu- tion with 336 × 560 × 560 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' A logarithmic grid (“ra- tioed” in ZEUS terminology) is employed, where cells become larger in each direction in a regular fashion as the distance from the origin increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' This allows us to cover a fairly large spatial region, while maintaining a reasonably high resolution in the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The x1 direction (perpendicular to the disk and parallel to the outflow) extends from 100 au above the disk midplane to 26, 500 au, while the x2 and x3 directions (parallel to the disk plane) extend out to ±16, 000 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Compared to the Paper I simulations, this domain is about twice as long in the x1 direction, and slightly larger in the x2 and x3 directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' All boundaries, except for the inner x1 boundary, are outflow boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The inner x1 boundary is more complicated, as the outflow is injected through it, and mass can “accrete” onto the disk through it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The fastest part of the disk wind is injected in a circular region with radius ri centered on the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Following Paper I, ri is related to the size of the disk around the protostar, rd (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2 of Paper I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Just outside of the injection region is a smoothing region, through which material is also injected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The role of this smoothing region is to gradually transition from the density and velocity of the injected disk wind profile to that of the surround- ing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The smoothing region has a radius of ro = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='8ri, somewhat larger than the value of ro = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='3ri used in Paper I to ensure that it contains several cells in the x2 and x3 directions at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Going further out is the accretion region, extending from ro to racc, through which material is removed to join the accretion disk at a controlled rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Beyond this, we use reflecting boundaries, to prevent any additional mass from flowing off the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The value of ri (and thus also ro) increases during the evolution as the star grows in mass, since rd ∝ m2/3 ∗ in the fiducial model of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2014b) in the limit of constant star formation efficiency, a fixed disk to star mass ratio, and a constant profile of rotational energy to gravitational energy ratio of material in the initial core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The radius of the accretion region, racc, adjusts over time so that the integrated mass flow rate through the annulus given by racc−ro that has outflow boundary conditions is ˙msim = 1 2 ˙m∗(1 + 1/3 + 1/10) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='72 ˙m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Note, the term 1/3 accounts for the growth of the accre- tion disk, which is assumed to have a mass md = m∗/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The term 1/10 is present to account for the injected mass flux of the disk wind that is immediately returned to the simulation grid through the injection region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The factor 1/2 is present since we simulate only one hemi- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The outer radius of the accretion region, racc, is adjusted so that the desired accretion rate is achieved via this region of outflow boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Initial core We initialize the simulation with a 1 M⊙ protostar located at the origin of our coordinate system, which is 100 au below the inner x1 boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' On the grid, we include one hemisphere of a 60 M⊙ core, with a radius of 12,000 au, which is the size expected for such a core embedded in a clump with mass surface density of Σcl = 1 g cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In the TCA model, the fiducial initial density struc- ture of the prestellar core is assumed to be spherical, with a power-law of the form ρ ∝ r−kρ with kρ = 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Thus our density structure is given by ρ(t = 0) = ρs (r/Rc)−3/2 , (1) 4 where ρs is the density at the surface of the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Note, in Paper I, which was mainly considering snapshots of later phases of the evolution, we adopted kρ = 1 as an approximation of the expected structure that develops in the expansion wave of the collapse solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' For our core with kρ = 3/2, we have ρs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5 × 10−18 g cm−3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', nH = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='1 × 106 cm−3 assuming a mass per H of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='34 × 10−24 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Beyond Rc we adopt a constant ambient density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='1ρs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The material in the core and its surroundings is initialized to be at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Following Paper I, the initial magnetic field configura- tion is the canonical Blandford & Payne (“BP”) config- uration (Blandford & Payne 1982), with a constant field added to it to ensure that the core flux is ∼ 1 mG × R2 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The BP configuration is a force-free, hour-glass shaped, purely poloidal magnetic field configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' At the mid-plane, the BP field varies as Bp ∝ r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The 1D velocity dispersion of the fiducial 60 M⊙ prestellar core, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', assuming virial equilibrium, is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='09(Mc/60 M⊙)1/4(Σcl/1 g cm−2)1/4 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In our simulation we adopt an isothermal equation of state with an effective sound speed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', signal speed, of cs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='90 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' This choice is made so that the core is moderately sub-virial and will undergo gravitational contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The gravitational field is treated with a simple ap- proximation in which the mass of the star and the disk, residing outside of the simulation domain, are treated as a point mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' For the contribution of the potential of the envelope material, we assume a simple model of a fixed core envelope size, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', of radius Rc, and a fixed power law index describing the radial distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', ρ ∝ r−3/2, but with the normalization of the profile adjusted to match the mass that is remaining in the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' When the simulation starts, the core immediately be- gins to contract as the initial setup is unstable to grav- itational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Initially, the plasma-β (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', where β ≡ Pgas/Pmag) is slightly above unity in the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' However, as the envelope collapses, the plasma-β drops below unity, meaning that the magnetic field starts to dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The collapse will therefore not be spherically- symmetric towards the protostar, but instead be guided along the field lines towards the mid-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Injection of the disk wind We launch the disk wind through the injection re- gion on the inner x1 boundary, with ˙minj = 1 2 1 10 ˙m∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='05 ˙m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We also enforce that the injected outflow has the same momentum rate in the x1 direction as in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2014b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Together, this can be used to constrain the injected density and velocity in the x1 direction (per- pendicular to the injection boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' As in Paper I, we then have an injected density: ρinj = � � � � � exp (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='0289 rcyl/r∗)φρρ0 rcyl < x0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='77 �rcyl x0 �−1 φρρ0 rcyl ≥ x0 (2) and an injected v1 velocity: vinj = (rcyl/r∗)−1/2φinjvK∗, (3) where r∗ is the stellar radius, x0 = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='3r∗, rcyl is the distance from the x1 axis, ρ0 is the injection density at the axis, vK∗ is the Keplerian speed on the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' φρ and φinj are time dependent dimensionless factors that are needed in order to obtain the desired mass flow and momentum rates of the inflowing wind, as is discussed in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The velocity components of the injected flow in the 2- and 3-directions are set so that the flow is along the direction of the initial magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The injected flow is also given an additional toroidal velocity compo- nent: vφ,inj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='23 � rcyl 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='4r∗ �−1/2 vK∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (4) The values employed for ri, ρ0, ˙m∗, ˙minj, and ˙pinj are given for protostellar masses of 1, 2, 4, 8, 16, and 24 M⊙ in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In the smoothing region, at ri < r < ro, the veloc- ity is gradually reduced by multiplying it by a factor w = cos2[ π 2 (r − ri)/(ro − ri)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The initial density of the surrounding envelope is gradually joined with the den- sity in the injection region by dividing the core density by 1+w(fjump−1), where fjump is the ratio of the initial core density to the density in the injection region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We note that, especially in the outflow cavity, if the density in a cell drops too low, the Alfv´en time step drops to such a low value that the simulation effectively grinds to a halt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' To avoid this, it is common practice in outflow simulations to implement a density floor, which prevents the Alfv´en time step from becoming extremely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' However, including such a density floor means mass is being artificially added to the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In this work, we have used a density floor that depends on height x1 above the disk: nH,floor = (x1/105 au)−1 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The rea- son for this choice is that near the inner x1 boundary where mass is accreting, we need a fairly large density floor to maintain a reasonable Alfv´en time step as the magnetic fields are stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' High above the disk, the density in the outflow cavity drops to values much below what the floor needs to be near the inner x1 boundary, and hence the density floor in the outer part of the sim- ulation box can be lower than in the inner part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Values of the radius of the injection region ri, the injected density along the axis ρ0, the desired accretion rate ˙macc, the desired injected mass flow rate ˙minj, and the desired injected momentum rate ˙pinj employed at the lower x1 boundary for protostellar masses m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' m∗ ri ρ0 ˙macc ˙minj ˙pinj [M⊙] [au] [10−17 g cm−3] [10−4M⊙ yr−1] [10−5M⊙ yr−1] [10−3M⊙ km s−1 yr−1] 1 92 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='9 2 106 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='9 4 124 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='4 8 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 16 196 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 24 282 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5 note that when mass is added to a cell in the simula- tion, we do not adjust the velocity of that cell, and as a consequence momentum is also added to the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Density, velocity and magnetic field structures We have simulated the evolution of the protostellar core for ∼ 105yr as the protostar grows from m∗ = 1M⊙ to about 26 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 1 we show slices of the density structure in the x1 − x2 plane at x3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' These images show the general structure of the disk-wind outflow cav- ity as it gradually carves open a larger and larger vol- ume from the initial core infall envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Concurrent with this evolution of the outflow cavity, we also see the collapse of the infall envelope down towards the central midplane base of the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' A movie showing the evo- lution of this structure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' During the course of the evolution the range of densities present in the simulation extends from nH ∼ 4cm−3 (in the outflow cavity) to ≳ 108 cm−3 (in the inner infall envelope).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Figure 3 shows the magnitude of the outflowing ve- locity along the x1 direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', v1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='9 km s−1, for the same slices through the simulation domain shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' At any given evolutionary stage, the highest velocities are found close to the central axis of the out- flow cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' At the earliest stages shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', m∗ = 2M⊙, these velocities are already ∼ 2, 000km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' By the later stages with m∗ = 24 M⊙, these velocities have risen to ∼ 5, 000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Figure 4 shows the magnitude of the total magnetic field strength for the same slices through the simula- tion domain shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The largest magnetic field strengths are ∼ 100mG near the base of the outflow and inner infall envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In the outflow cavity, the magnetic field strength is much lower than in the infall envelope, with values at low as ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='01 mG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Evolution of the outflow cavity opening angle To evaluate the opening angle of the outflow cavity at a given height x1, we first calculate the area A in the x2-x3 plane of the outflowing matter that has v1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='9 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We then approximate the outflow as having a conical shape with a circular cross section of area A = πr2, giving r = � A/π, and then find the half opening angle of that cone, tan(θoutflow) = r/x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 5 we show the evolution of the calculated opening angle over time for several different heights above the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' These direct estimates of the opening angles are stopped when the outflow cavity region approaches the lateral edges of the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Beyond this point, shown with dashed lines, we make an approximate estimate for opening angle at a given height via linear extrapolation from the closest lower height where the geometry of the outflow is still contained within the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' From our results we see that the outflow cavity open- ing angle is larger at lower heights (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', at 5,000 au), and is smaller at larger heights due to collimation of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In other words, the outflow cavity is not truly conical (as is evidenced in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Considering a fidcuial height equal to the initial radius of the core, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', 12,000 au, we see that the outflow cavity opening angle has achieved a value of about 10◦ at the earliest stages of the simulation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', when m∗ = 2 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' It then rises slowly until m∗ ∼ 4 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' After this it increases at a slightly faster rate, reach about 42◦ by the time m∗ = 18M⊙, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', the last stage where it can be directly evaluated in the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' An extrapolation based estimate at m∗ = 24 M⊙ yields θoutflow ≃ 50◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In Figure 5 we also compare our results to those of Paper I (without pre-clearing), which were calculated at the top of the grid in those simulations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', at a height of about 12, 000 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Recall that in Paper I, with models run at fixed m∗, it was somewhat uncertain at which time to evaluate the results for the opening angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Paper I also considered a case “with pre-clearing” that attempted to allow for the earlier stages of evolution and these yielded larger opening angles at the later stages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', about 50◦ at m∗ = 16M⊙ and 78◦ at 24M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We find that our new simulations with a continuous evolution followed from low to high values of m∗ yield moderately 6 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Slices of simulation results for density in the x1 − x2 plane at x3 = 0, with x1 corresponding to the outflow axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The top, middle and bottom rows show m∗ = 2 M⊙ and 4 M⊙, 8 M⊙ and 12 M⊙, and 16 M⊙ and 24 M⊙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' smaller cavity opening angles than the results of Paper I, with the biggest differences being at the highest masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We also compare our results to the opening angles predicted by the semi-analytic model of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2014b), following the method of Matzner & McKee (2000), which is based on the condition of whether the material in a given direction can be accelerated to the es- cape speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We find that our numerical results predict a moderately narrower outflow cavity geometry than this semi-analytic model, with the difference being about 20◦ by the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Mass and momentum fluxes of the outflow We evaluate the rate at which mass flows out of the top of the simulation box at the x2 − x3 boundary face via (1/2) ˙moutflow = � ρv1dA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', performing the sum- mation over the actual area of the outflow with no as- sumption of it being circular and equating this to half the total mass flux in a bipolar protostellar outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The evolution of this outflowing mass flux is shown in Fig- ure 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Initially, there is a transient phase with a fairly high mass flux out of the simulation box of ∼ 4×10−5M⊙yr−1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5×104 2×104 [np] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='93 x 104 5000 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='67 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 2x104 [np] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='40 104 5000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 t=94,000 yrs, M=24 2x104 [np] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='87 x 104 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='60 10000 0 10000 10000 0 10000 Log(n/[cm-3]) Lnp] X2 npl7 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Movie showing the temporal evolution of the x1 − x2 at x3 = 0 density slices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', same as the examples shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' while the outflow cavity is being cleared out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' After this the mass flow rate grows from about 2 × 10−5 M⊙ yr−1 to ∼ 1 × 10−4 M⊙ yr−1 by the time the star has reached ∼ 10M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We note that the mass flux exhibits moderate, ∼ 30%, fluctuations during this evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' After this the mass flux stops increasing and exhibits more dramatic fluctuations during the evolution to m∗ = 16 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' After this, it shows a more steady, smooth decline, which is mostly caused by the outflow cavity expanding beyond the size of the top face of the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' For this reason, we do not calculate the mass flow rate out of the grid for masses beyond ∼ 20 M⊙: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', at this stage a significant amount of mass is now leaving across the side boundaries (as can be observed in the movie in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2 and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Figure 6b shows the ratio of the mass flux leaving the top of the simulation domain to the mass injected at the base of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' After the initial peak asso- ciated with first breakout of the outflow, this ratio is about 2, but then rises up to a peak just below 10 when m∗ = 10M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' At higher masses it generally declines, but with large fluctuations, eventually reaching values near 2 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Figure 6c shows the time evolution of the total mass that has left the top of the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We find that more than 4 M⊙ has left the grid as part of the outflow by the time the protostar reaches 20 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Figure 7a shows the momentum flux passing through the top of the simulation domain, evaluated as ˙p = � ρv2 1dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 6, we cut off the measurements when substantial mass and momentum start to leave the domain through the side boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We find that the momentum flux leaving the domain stays approxi- mately constant at about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='005 M⊙ km s−1 yr−1, until the star reaches ∼ 7 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Then it increases to reach nearly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='02 M⊙ km s−1 yr−1 when the star is ∼ 16 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' It then continues to increase, but at a slower rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' How- ever, at this stage we begin to lose track of mass that is leaving through the sides of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Figure 7a also shows the injected momentum flux at the base of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In general, as expected, we see a very good agreement between the injected and ejected momentum fluxes, with the largest deviation occurring 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='93 2x104 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='67 [αu] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='40 +2 104 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='13 5000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104-1x104-5000 0 5000 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5×104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='60 x, [αu] 28 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Slices in the x1 − x2 plane at x3 = 0 of simulation results for total velocity, v, but only showing cells with v1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='9 km s−1 to highlight outflowing gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The top, middle and bottom rows show m∗ = 2 M⊙ and 4 M⊙, 8 M⊙ and 12 M⊙, and 16 M⊙ and 24 M⊙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' at late times due to some outflow material leaving via the sides of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The ratio of these momentum fluxes is shown explicitly in Figure 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Figure 7c shows the total momentum that has left via the top of the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' This grows steadily to reach ∼ 800M⊙ km s−1 by the time the protostar has reached ∼ 20 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Star formation efficiency Here we evaluate the star formation efficiency (SFE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', the ratio of the final stellar mass to the initial core mass, that is implied by our simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' After 100,000 years, the protostar has grown to m∗ ≃ 26 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Thus we estimate that ¯ϵ∗f ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' This is a lower limit since in our model the disk has a mass of mdisk = (1/3)m∗ ≃ 9 M⊙ and a significant portion of this ma- terial is expected to be able to accrete to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' If the only process diverting material from the accretion disk is injection into the disk wind with ˙mw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='1 ˙m∗, then the final stellar mass would be at least 34 M⊙, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', ¯ϵ∗f ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' It is possible that a larger fraction of ma- terial could be diverted from the accretion disk if other forms of feedback, especially disk photoevaporation, are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' However, Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2017) considered such models and found that disk photoevaporation was relatively unimportant compared to the disk wind mass flux for this mass and accretion rate regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The above estimates are likely to still be lower limits, since there is still 12M⊙ (3M⊙ from the initial core and 9 M⊙ from the surrounding clump) remaining in the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5×10 t=9,000 yrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' t=21,000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='70 [np] yrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' M=2 M= 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='95 5000 t=39,000 t=54,000 2x104 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='20 [no 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 yrs, yrs, M=8 M=12 5000 W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 t=68,000 yrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' t=94,000 2x104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 yrs, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='70 104 M=16 M=24 5000 @ 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='05 10000 0 10000 0 10000 X2 [np] X2 [au] log(v/[km s-1])9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Slices of simulation results for magnetic field strength, B, in the x1 − x2 plane at x3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The top, middle and bottom rows show m∗ = 2 M⊙ and 4 M⊙, 8 M⊙ and 12 M⊙, and 16 M⊙ and 24 M⊙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' simulation domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', 24 M⊙ in the global, mirrored domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' One expects that a significant fraction of this material would be accreted to the central protostar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In the case that all of the remaining initial core mass is accreted, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', 6 M⊙, then this would thus result in a SFE of ¯ϵ∗f ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Comparing the semi-analytic model of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2014b), they also reached a final value of m∗ = 26 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Thus, with the same considerations of residual disk ac- cretion, they expect to reach ¯ϵ∗f ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' However, their model at this point would be exhausted of gas and so this would be the final estimate of SFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Thus we con- clude that the expected SFE from our numerical model is moderately (∼ 20%) larger than that predicted by the semi-analytic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' This is consistent with the generally smaller outflow opening angles found during the course of the evolution in the numerical model com- pared the Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2014b) semi-analytic model (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' However, we note that in the fiducial TCA model of McKee & Tan (2003), the initial core is expected to in- teract with significant surrounding clump gas during its collapse to a protostar, so with this consideration the results of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2014b) for the final stellar mass, m∗f, are also lower limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' If SFE is defined with respect to the initial core mass, then the values of ¯ϵ∗f would also be lower limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 一 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='00 2x104 [np] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 104 5000 一 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x10* 2×104 [np] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5×104 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='00 104 5000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='00 [np] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 x 104 5000 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='00 10000 0 10000 10000 0 10000 X2 [np] ×2 [αu] Log(B/[G])10 0 10 20 30 40 50 60 70 80 0 5 10 15 20 25 θoutflow [degrees] m* [M⊙] Staff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2019) Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2014) height 5,000 au height 12,000 au height 20,000 au height 25,000 au 12,000 au extrapolated 20,000 au extrapolated 25,000 au extrapolated Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Outflow cavity opening angle measured at different heights above the disk (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Extrapolated estimates (dashed lines) are needed once the cavity nears the simulation boundary at a given height (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Also shown are the outflow cavity opening angles found in the numerical models of Paper I (squares) and the semi-analytic models of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2014b) (crosses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Outflow mass spectra One method of comparing our model results with ob- served systems is via the distribution of outflowing gas mass with line of sight velocity velocity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', “mass spec- tra”, since this can be inferred from observations of CO emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Note, in this paper we will not make synthetic CO spectra of our models, deferring this step to a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' To produce the distribution of mass with line of sight velocity, we need to produce a “global” simulation domain, which is achieved by mirroring our simulation grid about the x1 = 0 boundary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', the disk plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In this way we produce a symmetric bipolar outflow structure, which we then view at various angles, θview, to the outflow axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Note, θview = 0◦ is defined as a line of sight that is parallel to the outflow axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Figure 8 shows the mass spectra within the global do- main at various evolutionary stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Note, these spectra include all gas, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', both outflowing and infalling mate- rial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We have chosen three values of θview that are part of the grid of uniformly sampled grid of cos θview values in the radiative transfer models of Zhang & Tan (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The mass spectra show a sharp peak at low velocities, and, except for θview values close to 90◦, long tails to larger velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' As the protostellar mass increases, we find more mass at larger velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' For m∗ > 16 M⊙, the largest velocities are > 3000km s−1 when the system is viewed close to the outflow axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' One point to note is that between 2 M⊙ and 4 M⊙, the maximum velocities decrease somewhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' This is due to the protostellar ra- dius (which also sets the inner disk radius) growing from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='45 R⊙ at 2 M⊙ to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5 R⊙ at 4 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The injection ve- locity of the outflow is proportional to the Keplerian speed at the launching point (vKep ∝ m1/2 ∗ r−1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Hence, the highest velocity outflow is launched from the inner disk and, as the inner disk radius expands, the velocity of the material launched from the inner disc de- creases, even though the central mass is growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We use these mass spectra in the next subsection to make detailed comparisons to some observed massive proto- stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Comparison with observed outflow mass spectra In Figures 9 and 10 we compare the simulation out- flow mass spectra to equivalent outflow mass spectra of G35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='20-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='74N and G339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='88-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='26 (hereafter G35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 and G339) as derived from ALMA observations of CO(2- 1) line emission by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2022) and Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2019), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Note, the observed line emis- sion from these sources was extracted from regions of 11 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='4 0 5 10 15 20 1/2 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' outflow [10-4 M☉ yr-1] m* [M⊙] 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' outflow/m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' inj m* [M⊙] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5 0 5 10 15 20 ∫ 1/2 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' outflow dt [M☉] m* [M⊙] Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (a) Top: Evolution of outflow mass flux through the top of the simulation domain (x2 − x3 face at x1 = 25, 000 au) (purple solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The red dashed line shows the injected mass flow rate of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (b) Middle: Ratio of the mass flow rate out of the top of the simulation box to the injected mass flow rate at base of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (c) Top: Evolution of total mass that has left the top of the simulation domain by being swept-up by the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' ∼25,000 au in radial size centered on the protostars, similar to the size of our simulation box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We consider a velocity range of ±50 km s−1 and exclude the inner ±10 km s−1, which is affected by the presence of ambi- ent clump gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' To quantify the differences between the models and observations, we calculate the reduced χ2 between the two, following the method of Zhang & Tan (2018) (de- veloped for spectral energy distribution fitting), as: χ2 = 1 N � i �mi,data − mi,sim σ �2 , (5) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='025 0 5 10 15 20 1/2 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' outflow [M☉ km s-1 yr-1] m* [M⊙] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='8 0 5 10 15 20 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' outflow/p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' inj m* [M⊙] 0 100 200 300 400 500 600 700 800 900 0 5 10 15 20 ∫ 1/2 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' outflow dt [M☉ km s-1] m* [M⊙] Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (a) Top: Evolution of outflow momentum flux through the top of the simulation domain (x2 − x3 face at x1 = 25, 000 au) (purple solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The red dashed line shows the injected momentum flux at the base of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The green solid line shows the momentum flux injected in the semi-analytic model of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2014b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (b) Middle: Evolution of the ratio of the momentum flux through the top of the simulation domain to the injected momentum flux at the base of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (c) Bottom: Evolution of the total momentum that has left the top of the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' where N is the number of data points, mi,data and mi,sim are the mass in the i’th velocity bin in the observed data and in the simulation, and σ is the uncertainty on the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The uncertainty in the data is assumed to be comprised of a systematic uncertainty of 40% and a noise level that is ∼ 6 × 10−5 M⊙/(km s−1) (for both G35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 and G339).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Note that while the mass spectra are shown in log space, we perform the χ2 fitting in linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 12 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Distribution of outflow mass with line of sight velocity for material within a global (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', mirrored) simulation domain at various evolutionary stages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', protostellar masses) and as viewed at different inclination angles, θview = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='8◦, 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='4◦, 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='6◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' As seen in Figure 9, G35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2’s outflow mass spectrum at negative velocities is affected by a significant absorption feature at −20km s−1, which may be due to other molec- ular cloud components along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Thus, for this source we restrict fitting to only the positive veloc- ity range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Figure 10 shows that G339’s mass spectrum at positive velocities is similarly affected by absorption features and so here we only fit to the negative velocity range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Each of the panels in Figures 9 and 10 shows the mod- els at a particular evolutionary stage as seen over the full range of viewing angles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', uniformly sampling cosθview from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='025 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='975 in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We can see that at small values of m∗ the models generally fail to to match the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In particular, they underpredict the amount of outflowing gas at low and intermediate velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' For G35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2, there is a better agreement in the shape of the mass spectrum when m∗ ∼ 16M⊙ to 24M⊙, although the model is systematically low by a factor of about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' For G339, the shape of the mass spectrum has a best match when m∗ ∼ 20 M⊙, but is again low be about a factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We note that such systematic off- sets could be explained, at least in part, by uncertainties in the conversion of CO(2-1) line flux to mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The dif- ference could also simply be due to the observed systems being more massive protostellar cores, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', involving an initial core mass that is > 60 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Within the context of the Turbulent Core Accretion model, there is also the additional parameter of Σcl, which could be varied from the fiducial value of 1 g cm−2 assumed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Given the above considerations, we do not attempt to adjust our models further to find a better match to the data, since such a step will likely require running a much larger grid of simulations to explore the Mc and Σcl parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Nevertheless, with the context of the models we have presented, there is formally a best fitting model for each of G35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 and G339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' To illustrate these and the dependence of χ2 on model parameters, in Figure 11 we plot χ2 versus cos θview for all the consid- ered models at various evolutionary stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Again, we can see that the observations are more consistent with higher protostellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' However, in these higher mass cases, we note that the goodness of fit does not depend very sensitively on the viewing angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' c0s(0)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='025 (0=88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='69 cos(8)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='475 (8=61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='4)) cos(9)-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='975 (0-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='80) 2 M=2 M M=1E M 4 _ ) [ 2 4 M=8 8 2000 0 2000 2000 0 2000 v[km s v[krm s-]]13 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The mass velocity spectra from the simulation compared to that from observations of G35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='20-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='74N (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2022) for velocities less than ±50 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Comparison to other observational metrics of massive protostars The mass flow rate out of the simulation box (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 6) starts out at a few ×10−5 M⊙ yr−1 for the first ∼ 50, 000 years until the star reaches ∼ 10 M⊙, before increasing to more than 10−4 M⊙ yr−1 and becoming quite variable during the latter parts of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The momentum flux out of the simulation box (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 7) is, meanwhile, about 5 × 10−3 M⊙ km s−1 yr−1 for the first ∼ 40, 000 years until the star reaches ∼ 8 M⊙, after which the momentum rate grows steadily to ∼ 2 × 10−2 M⊙ km s−1 yr−1, and also shows time-variable behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Such values are in general agreement with observations of outflows from massive protostars (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Maud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Fedriani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2019), al- though it should be noted that there are significant un- certainties associated with the observational derivation of these mass and momentum fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' There have been a few measurements of magnetic field strengths in the outflows of massive protostars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In Orion Source I, which is thought to be 10 − 20 M⊙ protostar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', see discussion in Hirota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2020), the magnetic field strength was estimated to be 30 mG on a scale of a few hundred au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' This is in reasonable agreement with our simulations on similar scales (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' DISCUSSION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Comparison with previous simulation studies Here we discuss how our simulation results to those of other relevant studies of massive star formation, mostly restricting our consideration to those including pro- tostellar outflow feedback with magnetohydrodynamic (MHD) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The simulation we have presented, in addition to its initial core, has a well defined boundary condition during the evolution for the input protostel- lar outflow, which is tied to the evolution of the fidu- M=2 M M=16 M 2 4 1 M=20 2 4 M=24 2 4 M=12 Mo cos(0) 了 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='00 40 20 0 20 40 Ikm s14 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The mass velocity spectrum from the simulation compared to that from observations of G339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='88-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='25 (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2019), for velocities less than ±50 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' cial massive protostar in the Turbulent Core Accretion model (McKee & Tan 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2014b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' One comparable non-MHD simulation is that of Kuiper & Hosokawa (2018), who presented a simulation of a mas- sive protostar forming from a surrounding mass reservoir from 100 M⊙ to 1000 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The simulation code Pluto was utilized with a logarithmically spaced spherical co- ordinate grid assuming axial and midplane symmetry of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Feedback from radiation pressure, ion- ization and injected protostellar outflows was included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' However, the simulation did not include magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In contrast, the following simulation studies generally present collapse of a fully 3D gas structure to a sink particle representing a protostellar source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' For example, Rosen & Krumholz (2020) performed radiation MHD simulations of a collapsing 150 M⊙ core (significantly more massive than the 60 M⊙ core we consider in this study), and followed the evolution until the star reached a mass of 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='64 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' They found that once the stellar mass reached about 30 M⊙, radiation pressure created by the central star starts driving an expanding bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Radiative effects like this could potentially be relevant in our case if we continued the simulation beyond 30M⊙ (see also Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Commer¸con et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2021) compared collapse simula- tions of a 100 M⊙ core in several scenarios: without magnetic fields, with ideal MHD, and with ambipolar diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In the case of the non-magnetized simulation, they found a very weak outflow dominated by episodes of accretion bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In their ideal MHD simulation, they found that an increased pressure in the central region, due to increased stellar luminosity and build-up of mag- netic field, causes the outflow to almost disappear when the protostar reaches ∼ 10M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' However, this behaviour is not observed in their non-ideal MHD simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 1 M=2 M= 16 M 2 4 1 M=4 M M=20 M + 4 1 M=8 M M=24 3 4 1 2 cos(0) 4 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='00 40 20 0 20 40 [km s15 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Dependence of χ2 derived from fitting our sim- ulated mass spectra for different evolutionary stages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', various values of m∗) to the observational data of massive protostars G35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 (top) and G339 (bottom) as a function of the cosine of the viewing angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Mignon-Risse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2021b,a) performed radiation MHD collapse simulations also of a 100 M⊙ core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Mignon-Risse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2021a) focused on the out- flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' They found mass outflow rates of ∼ 10−5 − 10−4 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The momentum rate that they found was ∼ 10−4M⊙km s−1 yr−1, which is much smaller than the ∼ 10−3 − 10−2 M⊙ km s−1 yr−1 that we measure in our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We also note that our model involves the momentum rate growing as the protostellar mass grows, while they found a roughly constant momentum rate with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Also, contrary to our work, the opening angle in their simulations for the most part decreased with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The role of the magnetic field In ideal MHD, the gas is forced to follow the field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' This therefore creates a natural separation between the outflowing gas and the collapsing envelope, because the field lines found in the outflow are anchored in the in- jection region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' To demonstrate this we performed a test simulation with the same set up, but without magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 12, we show slices of the density structures and velocity fields of the outflowing gas for simulations with and without magnetic field after 39,000 years (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', when the protostar has reached 8 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' A consequence of the lack of magnetic field is less collimated, slower outflow, which interacts with much more envelope ma- terial, causing a larger mass flow rate out of the simu- lation box as more envelope material is entrained in the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We also find that the outflow cavity is much less distinct, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', in its density contrast with the infall envelope, in the simulation without magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Be- cause of this, there is no high-velocity outflow, and the momentum flow rate at a height of 25,000 au is smaller than in the simulation with magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Interestingly, the outflow pushes more material sideways when there is no magnetic field to confine it, forcing envelope material farther away from the protostar where the gravitational force is weaker, causing the envelope to collapse more slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' As a consequence, the envelope “puffs up” side- ways in the no-magnetic field simulation, and at 39,000 years it extends beyond the side boundaries (see density panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Effect of numerical resolution To examine the dependence on numerical resolution, we ran the same simulation set up with twice as many cells in each direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', 336×560×560 cells;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='1), but keeping other parameters the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In this higher resolution simulation, the smallest cells are now roughly 6 au on each side, compared to roughly 12 au in our primary “medium” resolution simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' This higher resolution simulation is much more computationally ex- pensive, and it was not feasible to run it for the entire evolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', up to ∼ 24M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Instead, we compare the results between the two resolutions at t = 39, 000 years, when the star has reached 8M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 13, we compare the logarithm of the number density, and the velocity field of the outflowing gas (where v1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='9 km s−1), in a slice through the middle of the grid (x3 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The medium and high resolution simulations are qual- itatively and quantitatively similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' For example, the opening angle of the outflow in the high resolution simu- lation measured at 12,000 au is 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='0◦, compared to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='0◦ in the medium resolution simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Note, while the low density part of the outflow cavity appears slightly larger in the slice of the high resolution simulation shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 13, the cavity defined by the outflowing gas is in fact slightly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' At 39,000 years, in the high resolu- tion simulation we find that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5 M⊙ has left the simula- tion box with the outflow through the outer x1 bound- ary, while in the medium resolution simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 M⊙ has left the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' These example diagnostics indicates a fairly good agreement between the higher and medium resolution simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 6 M: M M= 5 M= M: 20 24 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 3 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='8 1 cos(0)6 M: 248 M: M= M= 5 M= M: 20 24 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 3 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='8 1 cos(0)16 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The effect of magnetic fields on the outflow structure is illustrated by a comparison of the number density in the x1 − x2 slice at x3 = 0 and time 39,000 years, when the protostar is 8 M⊙ for a case without magnetic field (|B| = 0) (left panels) and with a magnetic field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', our fiduical model) (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The upper panels show density structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' the lower panels show the velocity field of the outflowing gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Log(n/[cm-3]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='67 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='93 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='20 time=39,000 yeurs Without B-field Medium resolution 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 2x104 [np] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 x 104 5000 10000 0 10000 10000 0 10000 X [au] X Inplog(v/[km s-1]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='67 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='35 time=39,000 ye0rs Without B-field with B-field 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x10 2x104 x 104 5000 10000 0 10000 10000 0 10000 X> [nD17 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Effect of numerical resolution is illustrated by a comparison of the density structure in the x1 − x2 plane at x3 = 0 at 39,000 years (m∗ = 8 M⊙) for the high resolution simulation (left panels) and fiducial medium resolution simulation (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The upper panels show density structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' the lower panels show the velocity field of the outflowing gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' CONCLUSIONS We have presented a 3D-MHD simulation of a magnetically-powered disk wind outflow from a massive protostar located at the center of a core with initial mass of 60 M⊙ and radius of 12,000 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Such a core is the fiducial case of the Turbulent Core Accretion model of McKee & Tan (2003), which involves the core being pres- sure confined by an ambient clump medium with mass surface density of Σcl = 1 g cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We have followed the evolution for 100,000 years as the protostar grows from m∗ = 1 M⊙ to about 26 M⊙, following the protostellar evolutionary track of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2014b), which sets both the accretion rate to the star and the mass and momentum injection rate to the disk wind outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We find that the protostar drives a powerful, colli- mated outflow that breaks out of the core at relatively early times, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', within ∼ 1, 000 yr of the start of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' At the scale of the initial core, the outflow has an opening angle (from outflow axis to cavity edge) of just over 10◦ until m∗ = 4 M⊙ at 21,000 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' There- after, as the protostar grows in mass and contracts to- Log(n/[cm-3]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='67 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='93 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='20 time=39,000 yeurs High resolution Medium resolution 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 2×104 [nD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x104 x 104 5000 10000 0 10000 10000 0 10000 Lau] X2 Inplog(v/[km s-1]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='67 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='35 time=39,000 yeurs High resolution Medium resolution 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5x10 2x104 x 104 5000 10000 0 10000 10000 0 10000 Lau] X> au18 wards the zero age main sequence, the outflow becomes more powerful causing the cavity to open up gradually, reaching opening angles of about 50◦ by the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' This disk wind outflow feedback thus dra- matically affects the density structure and morphology of the protostar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' While we have not performed radia- tive transfer (RT) calculations on these simulations (de- ferring this step for a future work), the RT models of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' (2014b) based on a semi-analytic core and outflow structure already illustrate the importance of such cavities for determining the infrared images and SEDs of the protostars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The outflow also is the main factor determining the star formation efficiency (SFE) from the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We find a lower limit to this SFE of ¯ϵ∗f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='43, but, considering the presence of a massive accretion disk and residual infall envelope, we estimate that the final value could reach as high as ¯ϵ∗f ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Such values are moderately higher than the efficiencies assumed of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='5 in the fiducial TCA model of McKee & Tan (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Inside the outflow cavity we find that the magnetic field is relatively weak, ∼ 10−4−10−5 G, while it retains its initial core value ∼ 10−3 G just outside the outflow cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Near the base of the outflow, however, we find magnetic field strengths of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='1 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The magnetic field structure we have implemented acts to help separate the outflow from the collapsing core, limiting the amount of the envelope material being entrained in the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The mass flow and momentum rates of our simu- lation are ∼ 2 × 10−5 − 2 × 10−4 M⊙ yr−1 and ∼ 2 × 10−3 − 2 × 10−2 M⊙ km s−1 yr−1 respectively, with these values controlled by the boundary conditions we have implemented, but also comparable to rates mea- sured from observed massive protostars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We have also compared the distribution of outflow mass with veloc- ity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', outflow mass spectra, of our simulations out to velocities of ±50 km s−1 with two example massive protostars G35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='2 and G339 observed by ALMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' This comparison indicates that such observations have di- agnostic power to constrain model parameters related to evolutionary stage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', m∗, and viewing angle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', θview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' While precise agreement between model and ob- servation is not found (and is not expected given po- tential systematic uncertainties in measure mass from CO line emission and from the limited range of TCA model parameters explored in our simulation), we do find quite striking agreement in the shape of the out- flow mass spectra for some models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' Further diagnostic tests involving full synthetic position-position-velocity cubes of synthetic CO line emission will be presented in a follow-up paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' JES, JPR and JCT acknowledge support from Collab- orative NSF grant AST-1910675.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' JES also acknowledges support from NASA through grant HST-AR-15053 from the Space Telescope Science Institute, which is operated by AURA, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=', under NASA contract NAS 5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' JPR also acknowledges support from the Virginia Ini- tiative on Cosmic Origins (VICO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' JCT also acknowl- edges support from ERC Advanced Grant MSTAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' We acknowledge the use of NASA High-End Comput- ing (HEC) resources through the NASA Advanced Su- percomputing (NAS) division at Ames Research Cen- ter to support this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQf2PnW/content/2301.00749v1.pdf'} +page_content=' The analysis and the fig- ures have been made using GDL (Coulais et al.' metadata={'source': 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Bally1,2,a, G. Giacalone3,b, M. Bender4,c +1 ESNT, IRFU, CEA, Universit´e Paris-Saclay, 91191 Gif-sur-Yvette, France +2 Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain +3 Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany +4 Universit´e de Lyon, Universit´e Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, 4 rue Enrico Fermi, F-69622, +Villeurbanne, France +Received: January 9, 2023 / Revised version: date +Abstract Having a detailed theoretical knowledge of +the low-energy structure of the heavy odd-mass nucleus +197Au is of prime interest as the structure of this isotope +represents an important input to theoretical simulations +of collider experiments involving gold ions performed +worldwide at relativistic energies. In the present article, +therefore, we report on new results on the structure +of 197Au obtained from state-of-the-art multi-reference +energy density functional (MR-EDF) calculations. Our +MR-EDF calculations were realized using the Skyrme- +type pseudo-potential SLyMR1, and include beyond +mean-field correlations through the mixing, in the spirit +of the Generator Coordinate Method (GCM), of particle- +number and angular-momentum projected triaxially de- +formed Bogoliubov quasi-particle states. Comparison +with experimental data shows that the model gives a +reasonable description of 197Au with in particular a +good agreement for most of the spectroscopic proper- +ties of the 3/2+ +1 ground state. From the collective wave +function of the correlated state, we compute an average +deformation ¯β(3/2+ +1 ) = 0.13 and ¯γ(3/2+ +1 ) = 40◦ for the +ground state. We use this result to construct an intrinsic +shape of 197Au representing a microscopically-motivated +input for precision simulations of the associated collider +processes. We discuss, in particular, how the triaxial- +ity of this nucleus is expected to impact 197Au+197Au +collision experiments at ultrarelativistic energy. +1 Introduction +For millennia, gold has held a prominent role in human +societies, whether it be as a symbol of wealth, a stan- +aE-mail: benjamin.bally@cea.fr +bE-mail: giacalone@thphys.uni-heidelberg.de +cE-mail: bender@ip2i.in2p3.fr +dard in international economic trades or because of its +medicinal and industrial applications. Interestingly, all +the gold of the world, whether it is used as jewelry, in +computer chips or kept in secured bank vaults, shares +one important feature: it is made of a single isotope. +Indeed, zooming in on the structure of this special el- +ement at the nuclear scale, one discovers that there is +only one stable gold isotope known to exist, namely +197Au. +As a matter of fact, nuclear physics essentially began +with the 197Au nucleus, which has been the first to be +discovered in 1909 by Rutherford, Geiger and Mardsen +from the scattering of α particles off a gold foil [1,2]. Over +100 years later, we have now a wealth of data available +on the structure of 197Au [3–10]. The low-energy spec- +trum of the nucleus is well known and electromagnetic +moments were measured for the ground state as well as +for several excited states [11,12]. Within a simple single- +particle picture, the 3/2+ +1 ground state of 197Au can be +interpreted as a proton 2d3/2 particle (hole) coupled to +a 196Pt (198Hg) core. Considering the naive picture of +a many-body state built as the product of independent +harmonic oscillator single-particles (holes) on top of a +suitably chosen core, oblate deformations are favoured +for nuclei close to the end of a major shell [13,14]. Given +the proximity of the Z = 82 and N = 126 shell closures, +we can thus expect 197Au to adopt a small oblate-like +deformation. Actually, axially-symmetric mean-field cal- +culations based on the Gogny D1S functional [15, 16] +reported in the AMEDEE database [17] do find an oblate +minimum with a magnitude of β ≈ 0.12. +Starting from the early 2000’s, 197Au has played a cen- +tral role as well in high-energy nuclear physics. Indeed, +gold ions are employed in various scattering experiments +ranging from fixed-target experiments at a nucleon- +arXiv:2301.02420v1 [nucl-th] 6 Jan 2023 + +2 +nucleon center-of-mass energy of 2-3 GeV performed +at GSI, Darmstadt, to ultrarelativsitic collisions at a +nucleon-nucleon center-of-mass energy of 200 GeV per- +formed at the at the BNL Relativsitic Heavy Ion Collider +(RHIC). Gold is, in particular, the prime species used at +the BNL RHIC, and the first conclusive evidence of the +formation of quark-gluon plasma in a laboratory has +been indeed obtained in ultrarelativistic 197Au+197Au +collisions [18–21]. +The theoretical interpretation of the results of high- +energy scattering experiments starts with an input from +nuclear structure theory [22]. The great success of the +hydrodynamic modeling of the quark-gluon plasma [23] +combined with the availability of data from collisions of +several ion species has recently lead to a precise identifi- +cation of the impact of the structural properties of the +collided nuclei on several experimental observables. In +particular, the azimuthal distributions of particles pro- +duced in relativistic collision experiments are observed +to present a strong sensitivity to spatial correlations of +nucleons (i.e. deformations) in the ground-state many- +body wave function of the colliding species [24–29]. For +example, in a recent article [30], we argued that we +could identify fingerprints of the triaxiality of 129Xe +in collisions performed at the CERN Large Hadron +Collider (LHC). The picture of a triaxial 129Xe drawn +from the analysis of high-energy data [31] is in excel- +lent agreement with results obtained from low-energy +Coulomb excitation experiments performed on the ad- +jacent isotopes, 128,130Xe [32, 33], as well as with our +recent theoretical calculations dedicated to these three +xenon isotopes [34]. Our goal for this manuscript is, in +a sense, to perform a similar analysis focused on 197Au, +to assess and potentially improve the current structure +input to high-energy 197Au+197Au collisions. +To this aim, we first investigate the low-energy structure +of 197Au on microscopic grounds using the MR-EDF +formalism [35,36]. More precisely, we present new results +obtained from state-of-the-art calculations based on the +configuration mixing of symmetry-projected triaxially +deformed Bogoliubov quasi-particle states [37–42] and +the use of the Skyrme-type pseudo-potential SLyMR1 +[43,44]. Secondly, we employ these results to construct a +point-nucleon density for 197Au, which we subsequently +employ in state-of-the-art simulations of the initial states +of high-energy 197Au+197Au collisions. We point out, +thus, the expected consequences of implementing our +newly-derived nucleon density in future hydrodynamic +simulations of such processes, with a focus on the role +played by the presence of a slight triaxiality in the +colliding ions. +This article is organized as follows: In Sec. 2, we re- +port on MR-EDF calculations dedicated to the study of +the structure of 197Au. Then, in Sec. 3 we analyze the +consequences of our results on the modeling and the ob- +servables of relativistic heavy-ion collisions. Finally, our +conclusions and prospects are reported in Sec. 4. +2 Nuclear structure +2.1 Method +In the present study, we use the same theoretical frame- +work as the one that was presented in Ref. [34] and +refer to that article for more details on our method +such as the definitions of the usual operators or the +symmetries used in our calculations. Nevertheless, to +deal with the heavy-mass 197Au nucleus we changed a +few numerical parameters compared to the ones used +in Ref. [34]. Firstly, the Bogoliubov reference states +were represented on a three-dimensional Cartesian La- +grange mesh [45] in a box of 32 points in each direction. +Secondly, when exploring the triaxial deformations, we +used a mesh with a spacing1 ∆q1 = ∆q2 = 375 fm2 +starting from (q1, q2) = (0, 0) and restricting ourselves +to positive values of q1 and q2, which maps the first +sextant of the β-γ plane. Finally, concerning the cutoffs +applied during the mixing of reference states: before the +mixing, we remove the projected components that in +the decomposition of the original reference states have +a weight that is lower than 10−3, whereas during the +mixing of K-components (performed individually for +each reference state) we remove the norm eigenstates +with an eigenvalue smaller than 10−2, and during the +final diagonalization mixing projected states originating +from different Bogoliubov vacua, we remove the norm +eigenstates with an eigenvalue smaller than 10−4 for all +nuclei. The values for the the cutoffs are more restric- +tive than the ones used when tackling the 128,129,130Xe +isotopes because the configuration mixing performed in +the present calculations for 197Au proved to be more +sensitive to the inclusion of components with a small +weight that are probably not well represented on our +cartesian mesh and, therefore, have to be discarded. Un- +fortunately, the improvement of the numerical accuracy +of our lattice, by increasing the number of mesh points +and/or reducing the spacing between them, implies a +substantial increase of the computational cost of the +MR-EDF calculations that is at present out of reach for +us. +1When considering axial deformations, this corresponds to a +step of ∆β ≈ 0.05. + +3 +Fig. 1: Particle-number restored total energy surfaces for +197Au and π = +1 (top panel) or π = −1 (bottom panel). +Black lines are separated by 1 MeV. The minimum for +positive (negative) parity, indicated by a silver star, +is located at a deformation of β = 0.12 and γ = 38◦ +(β = 0.12 and γ = 19◦) +2.2 Structure of 197Au +2.2.1 Energy surfaces +The first step in our approach is the generation of a set +of one-quasi-particle states that will be used as reference +states in the final configuration mixing calculations. To +generate and select the reference states, we follow the +strategy detailed in Ref. [34]. We briefly recall here +that this implies: i) the self-consistent blocking of four +different one-quasi-particle states at each point of the +deformation mesh, ii) the projection onto good particle +numbers and good angular momentum of all (converged) +one-quasi-particle states, and iii) the selection of the ones +having a projected energy lower than a given threshold +above the projected minimum of same parity. In this +work, we use a threshold of 5 MeV for both positive and +negative parity states. +But before discussing the final results obtained after con- +figuration mixing, let us first analyze the intermediate +steps in our method. Figure 1 displays the particle- +number restored (PNR) total energy surface for the +positive and negative parity states of 197Au. As can be +seen, the two energy surfaces exhibit a γ-soft topography +with a slightly deformed minimum2 located at β = 0.12. +Also, we notice that the surface for positive parity is +softer at small deformation than the surface for negative +parity. Finally, the minimum for positive parity states is +approximately 200 keV lower than the one for negative +parity states. +Performing the full symmetry restoration, we display +in Fig. 2 the angular-momentum and particle-number +restored (AMPNR) total energy surfaces for the lowest +Jπ = 1/2+, 3/2+ and 11/2− projected states, which +are the three values of Jπ giving the lowest projected +energies. A first remark is that the energy surfaces are +much more rigid with now a well pronounced triaxial +minimum with β = 0.13. Compared to the PNR case, +the minima of the AMPNR surfaces gain rouhgly 5 MeV +in binding energy and the absolute minimum is obtained +for Jπ = 3/2+. It is also worth mentioning that the one- +quasi-particle state giving the lowest projected state is +obtained by blocking a quasi-particle that is dominated +by a single-particle state originating from the spherical +2d3/2 shell. The latter observations are consistent with +the experimental spin-parity assignment 3/2+ +1 for the +ground state of 197Au as well as its naive single-particle +interpretation. However, we notice that the minimum +for the Jπ = 3/2+ surface is located at a deformation +with an angle γ = 24◦, which seems to be at variance +with the oblate-like shape expected from simple argu- +ments as mentioned above. Nevertheless, it is important +to remark that the configuration mixing may change +this picture. In addition, we displayed in Fig. 2 only +the surface for the lowest Jπ = 3/2+ projected states, +but given the fact that we explore triaxial deformations, +all the reference states with a non-zero average value +of γ will generate after angular-momentum restoration +two projected states with Jπ = 3/2+ that will enter the +configuration mixing. Ultimately, given the fact that the +AMPNR is only an intermediate step in our approach, +it is neither possible nor desirable to definitively charac- +terize the structure of the final correlated state at this +level of approximation. +Additionally, we note that the angular-momentum pro- +jection does not shift the energy minimum towards larger +values of β compared to the plain PNR case, which is +2Note that all the extrema discussed in this article are com- +puted from an interpolation based on the results obtained at +the points on the discretized deformation mesh. + +4 +Jπ = 1/2+ +Jπ = 3/2+ +Jπ = 11/2− +Fig. 2: Angular-momentum and particle-number re- +stored total energy surface for 197Au and for the lowest +Jπ = 1/2+ (top panel), the lowest Jπ = 3/2+ (mid- +dle panel) and lowest Jπ = 11/2− (bottom panel). +Black lines are separated by 1 MeV. The minima for +Jπ = 1/2+, 3/2+ and 11/2−, indicated by silver stars, +are located at deformations of β = 0.13 and γ = 39◦, +β = 0.13 and γ = 24◦ and β = 0.13 and γ = 22◦, +respectively +contrary to what is often observed in MR-EDF calcula- +tions [30,37–40]. +Fig. 3: Low-energy spectrum for 197Au. Experimental +data are taken from [46], which are based on the evalu- +ation [3] +2.2.2 Low-energy spectroscopy +Finally, we perform the full configuration mixing of sym- +metry projected reference states considering for positive +(negative) parity a set containing 24 (19) one-quasi- +particle states. In Fig. 3 we compare the theoretical +results to the available experimental data for the low- +lying states up to 1 MeV of excitation energies. First +of all, we remark that the theory is able to reproduce +the spin-parity assignment for the ground state (3/2+ +1 ) +as well as for the the first (1/2+ +1 ), second (3/2+ +2 ) and +third (5/2+ +1 ) excited states. As in experimental data, +the theory predicts a staggering between the two fomer +and two latter states but the relative spacing between +the two pairs of levels, as well as the spacing between +the levels within a pair, are too large. The 5/2+ +2 and +7/2+ +1 states also appear in our calculations but at too +high excitation energy. +The low-lying spectrum of positive parity states of 197Au +has been interpreted with De-Shalit’s core-excitation +model of odd-mass nuclei [47], within which an odd- +even nucleus is treated as a single nucleon coupled to +an even-even core. Whenever the excitation of the core +is energetically favored compared to the promotion of +the single nucleon to a higher orbital, in this model the +lowest lying excited states of the odd-even nucleus can +be interpreted as the single-particle configuration of the + +5 +ground state coupled in different ways to the lowest +excitation of the even-even core. When applied to 197Au +[6,48–51], the ground state of the nucleus is constructed +as a proton 2d3/2 particle (hole) coupled to a 196Pt +(198Hg) core with Jπ = 0+.3 Then, the weak coupling of +the same 2d3/2 particle, or hole, to the Jπ = 2+ excited +state of the (appropriate) core generates a quartet of +state with Jπ = 1/2+, 3/2+, 5/2+, 7/2+ whose energy +centroid Ec = {� +J(2J + 1)E(Jπ)} / {� +J(2J + 1)} is +equal to the energy E(2+) of the excited core [52]. Using +the experimental excitation energies, we obtain Ec = +364.2 keV that is close to the values of E(2+) = 355.7 +keV and for 411.8 keV for 196Pt and 198Hg, respectively. +Computing the energy centroid within our approach, +we obtain the value Ec = 630.7 keV that is obviously +too large compared to the experimental one but should +be compared the theoretical values of E(2+) for the +neighboring even-even nuclei calculated within the same +theoretical framework, which is technically possible but +falls outside the scope of the present article. +The MR-EDF theory also correctly predicts the 11/2− +1 +state to be lowest state of negative parity but with an +excitation energy of 792 keV, about 400 keV too high +compared to the experimental value of 409 keV. This +is especially surprising given that the energy difference +between the AMPNR minima for Jπ = 3/2+ and 11/2− +has the correct order of magnitude as can be seen Figs. 2. +As a matter of fact, the minimum for Jπ = 11/2− has +roughly the same energy as the one for Jπ = 1/2+. +What happens is that during the configuration mixing, +the 11/2− +1 state does not gain nearly as much correla- +tion energy as the positive parity states and, therefore, +ends up at a too high excitation energy. It is not en- +tirely clear why the mixing is less important in this +case. It might be due to the deficiency of the effective +interaction but we can not exclude the possibility that +other factors may play a role. For example, we had to +use more restrictive values for the cutoffs before and +after K-mixing to remove states not well represented on +our Cartesian mesh. Therefore, it is possible that some +important components or non-diagonal matrix elements +suffer from numerical inaccuracy. Another possibility +is that our selection strategy for the one-quasi-particle +to self-consistent block at the mean-field level misses +some configurations with negative parity relevant in the +subsequent shape mixing. +In general, the theoretical spectrum is too spread in +energy, which is an often-encountered deficiency of MR- +EDF calculations based on reference states generated by +3We mention in passing that some authors argue that using a +198Hg core provides a better global description of experimental +data [6]. +Quantity +Experiment +Theory +E(3/2+ +1 ) +-1559.384 +-1556.044 +rrms(3/2+ +1 ) +5.4371(38) +5.389 +µ(1/2+ +1 ) ++0.416(3) ++0.01 +µ(3/2+ +1 ) ++0.1452(2) +-0.38 +µ(5/2+ +1 ) ++0.74(6) ++0.15 +µ(5/2+ +2 ) ++3.0(5) ++0.14 +µ(7/2+ +1 ) ++0.84(7) ++0.51 +µ(9/2+ +1 ) ++1.5(5) ++0.81 +µ(11/2− +1 ) +(+)5.96(9) ++6.87 +Qs(3/2+ +1 ) ++0.547(16) ++0.65 +Qs(11/2− +1 ) ++1.68(5) ++2.05 +Table 1: Spectroscopic quantities for the low-lying states +of 197Au: total energy E (MeV), root-mean-square (rms) +charge radius rrms (fm), magnetic dipole moments µ +(µN), and spectroscopic quadrupole moments Qs (eb). +Experimental data are taken from [11,12,55–57]. The +experimental error on the binding energy is much smaller +than the rounded value given here +a variation of the total energy without consideration for +the angular momentum of the trial states. Indeed, such +a variation tend to energetically favor the ground state. +This deficiency can be in principle corrected by adding +a constraint on the average angular momentum of the +trial states during the minimization and using the value +of the constraint as an additional generator coordinate. +Unfortunately, such calculations are computationally +expensive and very few practical applications exist [53, +54]. +In Table 1, we report spectroscopic quantities for some +of the low-lying states. First, we see that the calcu- +lations reproduce fairly well the binding energy and +root-mean-square charge radius of the ground state, +with a relative accuracy below 1%. The spectroscopic +quadrupole moments for the 3/2+ +1 and 11/2− +1 states +are also reasonably well described in spite of being +slightly too large. While we indicate in Table 1 the value +for spectroscopic quadrupole moment of the ground +state, Qs(3/2+ +1 ) = 0.547(16) eb, currently taken as +the accepted value in the compilation of Ref. [58], and +which was determined using muonic hyperfine measur- +ments [51], we remark that other values appear in the +literature that are slightly larger, i.e. 0.60 eb and 0.64 +eb in Ref. [59] and 0.59 eb in Ref. [60], and in better +agreement with the value of 0.65 eb obtained in our +calculations. +Concerning the magnetic moments, they are, overall, +poorly described. The values of most of them are sig- +nificantly underestimated in our calculations and the + +6 +Transition +Type +Experiment +Theory +1/2+ +1 → 3/2+ +1 +E2 +35(3) +45 +M1 +0.004 +0.019 +3/2+ +2 → 1/2+ +1 +E2 +18(3) +6 +M1 +0.089(9) +0.048 +3/2+ +3 → 1/2+ +1 +E2 +9 +M1 +0.34 +3/2+ +2 → 3/2+ +1 +E2 +18.5(19) +0.4 +M1 +< 0.001 +0.002 +3/2+ +3 → 3/2+ +1 +E2 +4 +M1 +0.02 +5/2+ +1 → 1/2+ +1 +E2 +14.4(17) +12 +5/2+ +1 → 3/2+ +1 +E2 +26(6) +30 +M1 +0.034(4) +0.065 +5/2+ +2 → 1/2+ +1 +E2 +7.6(23) +8 +5/2+ +2 → 3/2+ +1 +E2 +7(6) +0.4 +M1 +0.083(10) +< 0.001 +7/2+ +1 → 5/2+ +1 +E2 +0.18(7) +1 +M1 +0.012(1) +0.106 +7/2+ +1 → 3/2+ +1 +E2 +33(3) +38 +7/2+ +1 → 3/2+ +2 +E2 +6.8(20) +0.3 +7/2+ +1 → 3/2+ +3 +E2 +3 +7/2+ +2 → 3/2+ +2 +E2 +6(4) +22 +7/2+ +2 → 3/2+ +3 +E2 +2 +7/2+ +2 → 5/2+ +1 +E2 +21(6) +13 +M1 +0.175(23) +0.010 +9/2+ +1 → 7/2+ +1 +E2 +10(7) +10 +M1 +0.028(10) +0.047 +9/2+ +1 → 5/2+ +1 +E2 +41(5) +43 +Table 2: Reduced transition probabilities among the +low-lying state of 197Au given in Weisskopf units. Ex- +perimental data are taken from [46], which are based on +the evaluation [3] +moment of the ground state has even the wrong sign. +Surprisingly, the best (relative) agreement with exper- +imental data is obtained for the magnetic moment of +the 11/2− +1 state. A similar mediocre description of the +magnetic moments was already observed in our study of +the 128,129,130Xe nuclei and we refer to this article [34] +for a discussion of the large spectrum of possible reasons +for this problem that is faced by the vast majority of +EDF calculations of magnetic properties. +In Table 2, we compare the theoretical values for the +reduced transition probabilities B(E2) and B(M1) to +available experimental data. Concerning the E2 transi- +tions, the theory gives reasonable estimates for most of +the decays. In particular, all of the strong transitions, +i.e. 1/2+ +1 → 3/2+ +1 , 5/2+ +1 → 3/2+ +1 , 7/2+ +1 → 3/2+ +1 and +9/2+ +1 → 5/2+ +1 , are well described. More generally, the +hierarchy between the transitions seems to be respected, +i.e. strong (weak) experimental transitions tend to be +strong (weak) in our calculations. One notable excep- +tion are the transitions towards/from the 3/2+ +2 state +that are largely underestimated in our calculations. A +possible interpretation is that the 3/2+ +2 and 3/2+ +3 states +are inverted in our calculation compared to the experi- +mental spectrum. Indeed, in Table 2 we also report the +calculated transitions towards/from the 3/2+ +3 state that +are in better agreement with the experimental data for +the transitions towards/from 3/2+ +2 state. In the limit +case of the core-excitation model discussed above, the +reduced transition probabilities from the states of quar- +tet 1/2+ +1 , 3/2+ +2 , 5/2+ +1 , 7/2+ +1 to the 3/2+ +1 ground state +are supposed to be equal among each other and with +the B(E2 : 2+ +1 → 0+ +1 ) of the even-even core. Obviously, +these equalities are not verified exactly for experimen- +tal data but the values remain somewhat close.4 In +particular, within the same model, the electromagnetic +transition probabilities are very sensitive to the mixing +of the Jπ = 3/2+ intrinsic states [48], a problem that +might also be present in our approach. +Concerning the M1 transitions, the model performs +poorly and most of the probabilities are either widely +underestimated or widely overestimated. These lacking +results are consistent with the observation made above +on the magnetic moments. Again, this characteristic is +a deficiency found in many nuclear EDF calculations. +While the projection techniques used here are crucial for +the reliable and unambiguous comparison of calculated +and experimental data for magnetic properties, they do +by themselves not lead to a satisfying description of +data. We refer again to [34] for further discussion of this +issue. +2.2.3 Collective wave functions +We now turn our attention towards the analysis of the +collective wave functions gJπ +σ (β, γ) of the correlated +states as defined in Ref. [34]. We recall here that the +squared collective wave functions (scwf) is a quantity +that can be used to gauge the importance of a given +deformation in the correlated wave function obtained +in the final step of the MR-EDF calculations, with the +caveat that, strictly speaking, it cannot be interpreted as +a probability distribution due to the non-orthogonality +of the reference states in the set. +In Fig. 4, we display the scwf for several low-lying states +of 197Au. Interestingly, in all cases, the distribution +of the scwf squared is dominated by triaxial shapes, +4We mention that the B(E2 : 2+ +1 → 0+ +1 ) values are 40.6(20) +and 28.8(4) W.u. for 196Pt and 198Hg, respectively [46,61,62]. + +7 +Jπσ = 1/2+1 +Jπσ = 3/2+1 +Jπσ = 3/2+2 +Jπσ = 3/2+3 +Jπσ = 5/2+1 +Jπσ = 5/2+2 +Jπσ = 7/2+1 +Jπσ = 7/2+2 +Jπσ = 9/2+1 +Jπσ = 11/2− +1 +Fig. 4: Collective wave function squared for several low-lying states of 197Au with different values of Jπ +σ . Black lines +are separated by 10% of the (respective) maximum value indicated by a silver star +with a sharply peaked maximum that has a quadrupole +deformation of β ≃ 0.14. But depending on the value of +Jπ +σ , the maximum is either located at angle γ ≈ 40◦ or +γ ≈ 20◦. In particular, we remark that the scwf of the +3/2+ +1 and 3/2+ +2 states exhibit different behaviour with +the former being located closer to the oblate axis whereas +the latter favours the prolate side of the sextant. Also, +the scwf of the 3/2+ +3 state is very similar to the one of +the 3/2+ +1 state. This is interesting for two reasons. First, +it is contrary to what could have been expected looking +at the AMPNR energy surface for Jπ = 3/2+ in Fig. 2. +Second, this is consistent with the oblate-like behavior +expected in an independent-particle model for a nucleus +close to the Z = 82 and N = 126 shell closures. Still, it +is important to stress that non-axial deformations carry +a substantial percentage of the scwf. +Looking more closely at the scwfs of the positive parity +states, we can arrange them into three groups of similar +appearance: a) the states 1/2+ +1 , 3/2+ +1 , 3/2+ +3 and 7/2+ +1 +that have a scwf mostly located in the range 30◦ ≤ +γ ≤ 60◦ b) the states 3/2+ +2 , 5/2+ +2 and 7/2+ +2 that have a +scwf mostly located in the range 0◦ ≤ γ ≤ 30◦ and c) +the states 5/2+ +1 and 9/2+ +1 whose scwf are more evenly +distributed as a function of γ and seem a combination +of the cases a) and b). This is in good agreement with +the data for the reduced transition probabilities given +in Table 2. Indeed, the transitions between the states of +a given group have very large B(E2) values whereas the +transitions between the states belonging to a different +group are less likely. This is not a perfect rule, however, +because the 5/2+ +1 state has also an strong transition +towards the 3/2+ +1 ground state but not towards the 1/2+ +1 +excited state even if the scwfs of the two latter states +have similarities. To come back to the core-excitation +model analysis, the fact that the scwf of the 1/2+ +1 , 3/2+ +3 ,5 +5/2+ +1 and 7/2+ +1 excited states have a large overlap with +5Provided that we interpret the 3/2+ +2 and 3/2+ +3 states as being +inverted in our calculations compared to experimental data. + +8 +the scwf of the 3/2+ +1 ground state is consistent with the +interpretation of the quartet of positive parity state as +being weak coupling of the same single-particle state to +a collective even-even core with an angular momentum +of either Jπ = 0+ or 2+. +As a last comment, we remark that the scwf of the +11/2− +1 state has a narrower distribution than the other +ones displayed, which is consistent with the fact that +this state does not mix as much when diagonalizing the +Hamiltonian within the space spanned by the symmetry- +projected reference states. +2.2.4 Average deformation +Finally, following the strategy presented in our previous +article on xenon isotopes [34], we use the scwf to compute +deformation parameters for the 3/2+ +1 ground state of +197Au and obtain: an average elongation of ¯β(3/2+ +1 ) = +0.13, with a standard deviation of ∆β(3/2+ +1 ) = 0.03, +and an average angle of ¯γ(3/2+ +1 ) = 40◦, with a standard +deviation of ∆γ(3/2+ +1 ) = 15◦. This average deformation +is consistent with the distribution displayed in Fig. 4 +as the maximum is located at a deformation of β = +0.14 and γ = 41◦ but the distribution extends towards +smaller values of β and is more or less equally distributed +with respect to the γ = 40◦ axis. +Within the rigid rotor model, it is also possible to com- +pute a deformation βr for the 0+ +1 ground state of an +even-even nucleus using the experimental B(E2) values, +for more details see for example Refs. [34,63]. Comput- +ing βr for the even-even nuclei adjacent to 197Au one +obtains the value 0.13 for 196Pt and 0.11 198Hg. Our +average deformation ¯β(3/2+ +1 ) = 0.12 fits nicely between +these two values, although we have to mention that +the definitions of the two elongations are model depen- +dent such that this excellent agreement may be partly +accidental. +The results of axially-symmetric EDF calculations based +on the Gogny D1S parametrization [15,16] reported in +the AMEDEE database [17] indicate a sharp minimum +at a deformation of about β ≈ 0.12 for 197Au. This is +perfectly consistent with our estimate. The AMEDEE +database also reports average deformations obtained +from large-scale five-dimensional collective Hamiltonian +(5DCH) calculations of even-even nuclei throughout the +nuclear chart [64]. We recall here that the 5DCH can be +derived as an approximation to the full GCM performed +here [63]. While their definition for the average deforma- +tion differs from ours, we mention that for 196Pt (198Hg), +they obtain an average elongation of 0.135 (0.110), with +a standard deviation of 0.032 (0.030), and average angle +of 32◦ (31◦), with a standard deviation of 12◦ (12◦). If +the values for the elongation are consistent with our +result, the average angles differ slightly with the 5DCH +result indicating a deformation right at the center of +the triaxial plane, although the fluctuations are large +enough such that the results are compatible. +3 Heavy-ion collisions +As previously mentioned, knowing the structure of 197Au +is of particular relevance in the context of high-energy +nuclear experiments, as gold is the primary species col- +lided at the BNL RHIC. This section analyzes the con- +sequences of our results for model simulations of ultra- +relativistic 197Au+197Au collisions. +3.1 Woods-Saxon parameterization of the ground +state +Traditionally, simulations of high-energy nuclear col- +lisions take as input from nuclear structure a point- +nucleon density which is used to sample nucleon coordi- +nates and define an interaction region between two ions +on a collision-by-collision basis.6 The standard choice for +the nucleon density is that of a deformed Woods-Saxon +(WS) profile: +ρ(r, θ, φ) = +ρ0 +1 + e[r−R(θ,φ)]/a , +(1) +where r, θ, φ are the usual spherical coordinates, ρ0 is +the saturation density, a is the surface diffuseness and +R(θ, φ) is the nuclear radius parameterized as +R(θ, φ) = R0 +� +1 + βWS +2 +� +cos(γWS)Y20(θ, φ) +(2) ++ +√ +2 sin(γWS)Re +� +Y22(θ, φ) +�� ++ βWS +4 +Y40(θ, φ) +� +, +where the spherical harmonics Ylm(θ, φ) are in complex +form. Note that the shape parameters βWS +2 +, γWS and +βWS +4 +represent surface deformations that differ from +the volume deformation reported in the analysis of the +previous sections [69]. +We consider now the intrinsic shape of 197Au computed +from a single Hartree-Fock-Bogoliubov (HFB) calcula- +tion with the SLyMR1 interaction in which the expecta- +tion value of the quadrupole operators are constrained +6More sophisticated calculations based on nuclear configura- +tions obtained from ab initio nuclear theory have also been +recently performed [65–68]. For the moment, they are limited +to the description of collisions of 16O ions. + +9 +Parameter +Proton +Neutron +Nucleon +ρ0 +0.067 +0.090 +0.157 +R0 +6.44 +6.65 +6.56 +a +0.46 +0.49 +0.48 +βWS +2 +0.134 +0.137 +0.135 +γWS +43◦ +43◦ +43◦ +βWS +4 +-0.024 +-0.023 +-0.023 +Table 3: Parameters for the point-proton, point-neutron +and point-nucleon densities defined as in Eq. (1) and +fitted to reproduce the one-body densities of a quasi- +particle state constrained to have, on average, β = 0.13 +and γ = 40◦; see the body of the text for more details. +The parameters R0 and a are given in units of fm, +whereas ρ0 is given in units of fm−3 +Fig. 5: Schematic illustration of the shape of 197Au based +on the surface parametrization of the matter density of +Eq. (2), and using the parameters reported in Tab. 3 +such that the one-body density of the trial one-quasi- +particle state7 verifies, on average, β = ¯β(3/2+ +1 ) = 0.13 +and γ = ¯γ(3/2+ +1 ) = 40◦.8 We fit the resulting one- +body nucleon density with the Woods-Saxon profile +given in Eq. (1). The fit parameters are reported in +Tab. 3. We obtain, thus, a new microscopically mo- +tivated parametrization for the Woods-Saxon profile +representing the nucleon density of the ground state of +197Au which can be employed in simulations of high- +energy collisions. This profile corresponds to a triaxial +ellipsoid with radii 6.02 fm, 6.68 fm, and 6.97 fm, as +illustrated in Fig. 5. +7The trial one-quasi-particle state is built by blocking a single- +particle state originating from the spherical 2d3/2 shell. +8All other non-vanishing multipole moments authorized by +the symmetries of our calculations are let free to adopt a value +that minimizes the total energy of the trial quasi-particle +state. +For completeness, we evaluate as well the neutron skin +of the intrinsic shape, as defined by the difference of rms +radii, ∆rnp = ⟨r2⟩1/2 +n +−⟨r2⟩1/2 +p +. For the density returned +by the constrained HFB calculation, we find +∆rnp[HFB(¯β, ¯γ)] = 0.17 fm, +(3) +which is in perfect agreement with the result obtained +from the full MREDF calculation +∆rnp[MREDF] = 0.17 fm. +(4) +On the other hand, the fitted Woods-Saxon profile gives +a neutron skin +∆rnp[WS fit] = 0.19 fm, +(5) +meaning that, even for a large nucleus such as 197Au, the +Woods-Saxon parametrization does not fully capture +skin differences of order 0.1 fm between neutrons and +protons. We note that both the above estimates agree +with a recent measurement of the STAR collaboration +obtained via diffractive photo-production of ρ0 mesons +in ultra-peripheral 197Au+197Au collisions [70], +∆rnp[STAR] = 0.17 ± 0.03 (stat.) ± 0.08 (syst.) fm. (6) +We note, in addition, that the half-width radius obtained +for 197Au by the STAR collaboration, R0[STAR] = +6.53 ± 0.06 fm, is fully consistent with that exhibited by +our nucleon density, R0[WS fit] = 6.56 fm. This suggests +that the density of gluons relevant for scattering at +these beam energies is in fact very close to the rest- +frame point-nucleon density. This potentially adds to +the circumstantial evidence of a small nucleon width in +high-energy collisions mediated by gluons [71–75]. +We discuss now the observational consequences of our +newly-derived nucleon density for relativistic 197Au+197Au +collisions. Model calculations of such processes (see e.g. +Ref. [76] for a state-of-the-art Bayesian analysis) have +so far employed the charge density of the nucleus, as +inferred from low-energy electron-nucleus scattering ex- +periments [77], as a proxy for the nucleon density. The +corresponding radial profiles are R0 = 6.38 fm, and +a = 0.53 fm. Nuclear quadrupole deformation has been +instead included by simply implementing βWS +2 += −0.13, +as reported by finite-range liquid drop model evalua- +tions [78]. In terms of radial profiles, there are, thus, +minor differences between the WS parametrization that +we show in Tab. 3 and that implemented in the lit- +erature. We only note a reduction by 0.05 fm in the +diffuseness parameter, a, which is due to the inclusion +of the neutron density. This will have a mild, though +visible impact on the initial eccentricities, εn, of the sys- +tem [79–81]. A new feature of our calculation is instead +the fact that 197Au is not fully oblate, but presents +γWS = 43◦. We investigate now the impact of such a +feature on high-energy collisions. + +10 cm +100% +x1013 +6.02 fm +6.68 fm +6.97 fm10 +3.2 Impact of the triaxiality +In the context of multi-particle correlation measure- +ments in the soft sector of high-energy nuclear colli- +sions, the strongest sensitivity to the triaxial structure +of the colliding nuclei is carried by the mean momentum- +elliptic flow correlation [82–84], +ρ2 ≡ ρ(⟨pt⟩, v2 +2) = ⟨⟨pt⟩v2 +2⟩ − ⟨⟨pt⟩⟩⟨v2 +2⟩ +σ(⟨pt⟩)σ(⟨v2 +2⟩) +, +(7) +where outer brackets denote a statistical average over +events, and σ(o) is the standard deviation of observable +o. This quantity can be evaluated in the final states +as a three-particle correlation [85], and it measures +the strength of the statistical correlation between the +charged-particle average transverse momentum, ⟨pt⟩, +and the charged-particle elliptic flow, v2, at a given +collision multiplicity. +To assess the impact of γWS = 43◦ on the ρ2 correlator of +197Au+197Au collisions, we follow Ref. [30] and provide +an estimate of the measured ρ2 from high-statistics simu- +lations of the initial condition of these processes. For the +details of such simulations, we refer to the exhaustive de- +scriptions given in Ref. [30]. Briefly, we assume that the +distribution of final-state multiplicities is proportional +to the distribution of initial-state entropy, S, which we +calculate event-to-event following the original TRENTo +parametrization [86] (s(x, τ0) ∝ √TATB, S = +� +d2x s) +with a nucleon size w = 0.5 fm, and a fluctuation pa- +rameter, k, tuned to reproduce measured multiplicity +histograms in 208Pb+208Pb collisions at CERN LHC +energy. We consider that i) the mean transverse momen- +tum is, at a given entropy, proportional to the initial +E/S, where E is the total energy of the system [87,88], +obtained upon application of the equation of state of +high-temperature QCD (e(x) ∝ s(x)4/3, E = +� +d2x e), +and ii) that the elliptic flow is proportional to the initial +eccentricity of the system, ε2. The Pearson correlation +coefficient of Eq. (7) can then be estimated by replac- +ing v2 +2 and ⟨pt⟩ with, respectively, ε2 +2 and E/S. Note +that the resulting estimator should not be compared +directly to the experimental measurements, as it misses +effects related to the cuts in transverse momentum, pt, +implemented in the experimental analysis, which have +been shown to be sizable for the magnitude of this +observable [31,89,90]. That said, it is the initial-state +estimator that carries the dependence on the deforma- +tion parameters, such that the relative impact of the +value of γWS on the final-state result can be assessed +from it [30,91]. +We perform 20 × 106 minimum bias simulations of +197Au+197Au collisions for three structure scenarios, +300 +350 +400 +450 +500 +550 +600 +Nrec +ch (|η| < 0.5) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +ρ +� +⟨pt⟩, v2 +2 +� +← uncertainty on STAR data at Nrec +ch ≈ 550 +TRENTo, 200 GeV Au+Au +oblate gold (βWS +2 += 0.135, γWS = 60◦) +triaxial gold (βWS +2 += 0.135, γWS = 43◦) +prolate gold (βWS +2 += 0.135, γWS = 0) +16 +9 +3 +1 +centrality (%) +Fig. 6: Initial-state estimates of ρ(⟨pt⟩, v2 +2) in 200 GeV +197Au+197Au collisions for prolate ions (dot-dashed +line), oblate ions (dotted line) and triaxial ions (dashed +line) presenting γWS = 43◦, as a function of the number +of reconstructed charged tracks in the STAR detector. +Shaded bands (of the same width as the lines) are statis- +tical uncertainties. The figure reports as well the total +uncertainty on preliminary STAR measurements for this +observable at high multiplicities. +namely, we set βWS +2 += 0.135, and consider γWS = 0◦, +43◦, and 60◦.9 Rescaling the TRENTo entropy to match +the observed mutliplicity of reconstructed charged tracks +in the STAR detector, N rec +ch , at midrapidity (|η| < 0.5), +our results for ρ2 are reported in Fig. 6. Qualitatively, +the impact of γWS follows the generic parametric expec- +tation ρ2 ∝ c0 − c1(βWS +2 +)3 cos(3γWS), where c0 and c1 +are positive coefficients [30,91]. We conclude that a 17◦ +deviation from oblateness in 197Au leads to a correction +of order 10-15% to ρ2 for collisions in the 0-2% centrality +range. We reiterate that, while our results for the magni- +tude of the Pearson coefficient should not be compared +directly to data, we expect the correction induced by +the triaxiality, relative to the oblate scenario, to be ro- +bustly captured by our initial-state evaluation. In Fig. 6 +we report as well the size of the experimental error on +preliminary ρ2 data at high multiplicity from the STAR +collaboration [92]. The error bar turns out to be signifi- +cantly smaller than the splitting that we find between +the triaxial scenario (red dashed line) and the oblate +scenario (dotted blue line). Therefore, according to our +results the impact of the triaxiality has been already iso- +lated in the preliminary data, and it will be possible to +9We safely neglect the effect of the very small hexadecapolarity +of the nucleus, βWS +4 += −0.023, in these simulations. + +11 +quantify it in the future via high-precision hydrodynamic +simulations. We stress, though, that the most effective +way to access the value of γWS is by studying the ρ2 +correlator of 197Au+197Au collisions normalized with +that of 238U+238U collisions, as done in Refs. [30, 31] +to extract such an information in the comparisons of +129Xe+129Xe and 208Pb+208Pb collisions, which allows +one to fully cancel theoretical and experimental system- +atical uncertainties and isolate transparent information +about the nuclear structure. The current mismatch be- +tween hydrodynamic results and experimental data for +238U+238U collisions [93] prevents us, for the moment, +from performing such an analysis, which will be thus +reported in future work. +4 Conclusions +In the present article, we first reported on new re- +sults on the low-energy structure of the heavy odd- +mass nucleus 197Au obtained by performing state-of- +the-art MR-EDF calculations that include the mixing +of angular-momentum and particle-number projected +Bogoliubov quasi-particle states with different average +triaxial shapes. All the calculations were realized using +the parametrization SLyMR1 of a Skyrme-type pseudo- +potential [44,94]. +Although odd-mass nuclei represent half of the existing +nuclei in the nuclear chart, their calculations within +the full-fledged MR-EDF framework are still scarce, ex- +ceptions being [34,40,41,95]. In this work, to generate +reference states adapted to the modeling of odd-mass +nuclei, we performed self-consistent blocking of Bogoli- +ubov one-quasi-particle states and considered exactly +all the time-odd terms of the functional. +The results obtained on the low-energy spectroscopy +of 197Au are reasonable. The spin-parity assignments +for the 3/2+ +1 ground state and for the first few excited +states are correct even if the levels are too spread out, +a well-known deficiency of usual MR-EDF calculations +that can be corrected by adding a supplemental con- +straint on the average angular momentum of the trial +wave functions when generating the set of reference +states to be projected and mixed [53,54]. The binding +energy, root-mean-square charge radius and spectro- +scopic quadrupole moment of the of the ground state +are also well reproduced. By contrast, the calculations +fail to reproduce the known magnetic moments for the +ground and excited states. Concerning the electromag- +netic transitions, the values for the reduced transition +probabilities B(E2) are, overall, well described whereas +the values for the B(M1) are off, sometimes by more +than one order of magnitude. +Starting from the collective wave function of the ground +state, we computed average triaxial deformation param- +eters ¯β(3/2+ +1 ) = 0.13 and ¯γ(3/2+ +1 ) = 40◦. Following the +the strategy of Ref. [30], we then fitted the parameters +of a deformed Woods-Saxon density profile, to obtain +a new state-of-the-art microscopically-motivated input +for the simulation of high-energy 197Au+197Au colli- +sions. In terms of radial profile parameters, our result +corrects to some extent the widely- and incorrectly- +employed charge-density parametrization, which has in +particular a too large skin thickness. For future precision +phenomenological studies of 197Au+197Au collisions, es- +pecially in view of the upcoming sPHENIX program +at the BNL RHIC, it will be crucial to implement real- +istic properties of the point-nucleon density in Monte +Carlo simulations. This includes as well implementing +an appropriate triaxiality, of order 45◦, for gold ions. +Our estimates indicate that this magnitude of the tri- +axiality does impact the final state in a significant way, +and we expect future theoretical work to be able to +cleanly isolate such a contribution from the data. As +an outlook, we emphasize that measurements of the +third centered moment (skewness) of the distribution of +⟨pt⟩ [96] provide additional and independent information +about γWS [91], and can be used in conjunction with +hydrodynamic simulations to further test our prediction +for this parameter. +Acknowledgements We thank Chunjian Zhang for help with +the entropy-to-multiplicity conversion used in Fig. 6, and +Wouter Ryssens for useful discussions. This project has re- +ceived funding from the European Union’s Horizon 2020 re- +search and innovation programme under the Marie Sk�lodowska- +Curie grant agreement No. 839847. M.B. acknowledges sup- +port by the Agence Nationale de la Recherche, France, un- +der grant No. 19-CE31-0015-01 (NEWFUN). 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C 103 +(2021) 024910. doi:10.1103/PhysRevC.103.024910. +URL +https://link.aps.org/doi/10.1103/PhysRevC.103. +024910 + diff --git a/D9E0T4oBgHgl3EQfggHn/content/tmp_files/load_file.txt b/D9E0T4oBgHgl3EQfggHn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8144f72de37d7510c5a72b5b1a89b2ddbcffed18 --- /dev/null +++ b/D9E0T4oBgHgl3EQfggHn/content/tmp_files/load_file.txt @@ -0,0 +1,1918 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf,len=1917 +page_content='Noname manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' (will be inserted by the editor) The shape of gold B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Bally1,2,a, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Giacalone3,b, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Bender4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='c 1 ESNT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' IRFU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' CEA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Universit´e Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 91191 Gif-sur-Yvette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' France 2 Departamento de F´ısica Te´orica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Universidad Aut´onoma de Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' E-28049 Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Spain 3 Institut f¨ur Theoretische Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Universit¨at Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Philosophenweg 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' D-69120 Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Germany 4 Universit´e de Lyon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Universit´e Claude Bernard Lyon 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' CNRS/IN2P3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' IP2I Lyon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' UMR 5822,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 4 rue Enrico Fermi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' F-69622,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Villeurbanne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' France Received: January 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 2023 / Revised version: date Abstract Having a detailed theoretical knowledge of the low-energy structure of the heavy odd-mass nucleus 197Au is of prime interest as the structure of this isotope represents an important input to theoretical simulations of collider experiments involving gold ions performed worldwide at relativistic energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In the present article, therefore, we report on new results on the structure of 197Au obtained from state-of-the-art multi-reference energy density functional (MR-EDF) calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Our MR-EDF calculations were realized using the Skyrme- type pseudo-potential SLyMR1, and include beyond mean-field correlations through the mixing, in the spirit of the Generator Coordinate Method (GCM), of particle- number and angular-momentum projected triaxially de- formed Bogoliubov quasi-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Comparison with experimental data shows that the model gives a reasonable description of 197Au with in particular a good agreement for most of the spectroscopic proper- ties of the 3/2+ 1 ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' From the collective wave function of the correlated state, we compute an average deformation ¯β(3/2+ 1 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='13 and ¯γ(3/2+ 1 ) = 40◦ for the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We use this result to construct an intrinsic shape of 197Au representing a microscopically-motivated input for precision simulations of the associated collider processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We discuss, in particular, how the triaxial- ity of this nucleus is expected to impact 197Au+197Au collision experiments at ultrarelativistic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 1 Introduction For millennia, gold has held a prominent role in human societies, whether it be as a symbol of wealth, a stan- aE-mail: benjamin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='bally@cea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='fr bE-mail: giacalone@thphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='uni-heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='de cE-mail: bender@ip2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='in2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='fr dard in international economic trades or because of its medicinal and industrial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Interestingly, all the gold of the world, whether it is used as jewelry, in computer chips or kept in secured bank vaults, shares one important feature: it is made of a single isotope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Indeed, zooming in on the structure of this special el- ement at the nuclear scale, one discovers that there is only one stable gold isotope known to exist, namely 197Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' As a matter of fact, nuclear physics essentially began with the 197Au nucleus, which has been the first to be discovered in 1909 by Rutherford, Geiger and Mardsen from the scattering of α particles off a gold foil [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Over 100 years later, we have now a wealth of data available on the structure of 197Au [3–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The low-energy spec- trum of the nucleus is well known and electromagnetic moments were measured for the ground state as well as for several excited states [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Within a simple single- particle picture, the 3/2+ 1 ground state of 197Au can be interpreted as a proton 2d3/2 particle (hole) coupled to a 196Pt (198Hg) core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Considering the naive picture of a many-body state built as the product of independent harmonic oscillator single-particles (holes) on top of a suitably chosen core, oblate deformations are favoured for nuclei close to the end of a major shell [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Given the proximity of the Z = 82 and N = 126 shell closures, we can thus expect 197Au to adopt a small oblate-like deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Actually, axially-symmetric mean-field cal- culations based on the Gogny D1S functional [15, 16] reported in the AMEDEE database [17] do find an oblate minimum with a magnitude of β ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Starting from the early 2000’s, 197Au has played a cen- tral role as well in high-energy nuclear physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Indeed, gold ions are employed in various scattering experiments ranging from fixed-target experiments at a nucleon- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='02420v1 [nucl-th] 6 Jan 2023 2 nucleon center-of-mass energy of 2-3 GeV performed at GSI, Darmstadt, to ultrarelativsitic collisions at a nucleon-nucleon center-of-mass energy of 200 GeV per- formed at the at the BNL Relativsitic Heavy Ion Collider (RHIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Gold is, in particular, the prime species used at the BNL RHIC, and the first conclusive evidence of the formation of quark-gluon plasma in a laboratory has been indeed obtained in ultrarelativistic 197Au+197Au collisions [18–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The theoretical interpretation of the results of high- energy scattering experiments starts with an input from nuclear structure theory [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The great success of the hydrodynamic modeling of the quark-gluon plasma [23] combined with the availability of data from collisions of several ion species has recently lead to a precise identifi- cation of the impact of the structural properties of the collided nuclei on several experimental observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In particular, the azimuthal distributions of particles pro- duced in relativistic collision experiments are observed to present a strong sensitivity to spatial correlations of nucleons (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' deformations) in the ground-state many- body wave function of the colliding species [24–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' For example, in a recent article [30], we argued that we could identify fingerprints of the triaxiality of 129Xe in collisions performed at the CERN Large Hadron Collider (LHC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The picture of a triaxial 129Xe drawn from the analysis of high-energy data [31] is in excel- lent agreement with results obtained from low-energy Coulomb excitation experiments performed on the ad- jacent isotopes, 128,130Xe [32, 33], as well as with our recent theoretical calculations dedicated to these three xenon isotopes [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Our goal for this manuscript is, in a sense, to perform a similar analysis focused on 197Au, to assess and potentially improve the current structure input to high-energy 197Au+197Au collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' To this aim, we first investigate the low-energy structure of 197Au on microscopic grounds using the MR-EDF formalism [35,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' More precisely, we present new results obtained from state-of-the-art calculations based on the configuration mixing of symmetry-projected triaxially deformed Bogoliubov quasi-particle states [37–42] and the use of the Skyrme-type pseudo-potential SLyMR1 [43,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Secondly, we employ these results to construct a point-nucleon density for 197Au, which we subsequently employ in state-of-the-art simulations of the initial states of high-energy 197Au+197Au collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We point out, thus, the expected consequences of implementing our newly-derived nucleon density in future hydrodynamic simulations of such processes, with a focus on the role played by the presence of a slight triaxiality in the colliding ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This article is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 2, we re- port on MR-EDF calculations dedicated to the study of the structure of 197Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Then, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 3 we analyze the consequences of our results on the modeling and the ob- servables of relativistic heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Finally, our conclusions and prospects are reported in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 2 Nuclear structure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='1 Method In the present study, we use the same theoretical frame- work as the one that was presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' [34] and refer to that article for more details on our method such as the definitions of the usual operators or the symmetries used in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Nevertheless, to deal with the heavy-mass 197Au nucleus we changed a few numerical parameters compared to the ones used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Firstly, the Bogoliubov reference states were represented on a three-dimensional Cartesian La- grange mesh [45] in a box of 32 points in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Secondly, when exploring the triaxial deformations, we used a mesh with a spacing1 ∆q1 = ∆q2 = 375 fm2 starting from (q1, q2) = (0, 0) and restricting ourselves to positive values of q1 and q2, which maps the first sextant of the β-γ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' concerning the cutoffs applied during the mixing of reference states: before the mixing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' we remove the projected components that in the decomposition of the original reference states have a weight that is lower than 10−3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' whereas during the mixing of K-components (performed individually for each reference state) we remove the norm eigenstates with an eigenvalue smaller than 10−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' and during the final diagonalization mixing projected states originating from different Bogoliubov vacua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' we remove the norm eigenstates with an eigenvalue smaller than 10−4 for all nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The values for the the cutoffs are more restric- tive than the ones used when tackling the 128,129,130Xe isotopes because the configuration mixing performed in the present calculations for 197Au proved to be more sensitive to the inclusion of components with a small weight that are probably not well represented on our cartesian mesh and, therefore, have to be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Un- fortunately, the improvement of the numerical accuracy of our lattice, by increasing the number of mesh points and/or reducing the spacing between them, implies a substantial increase of the computational cost of the MR-EDF calculations that is at present out of reach for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 1When considering axial deformations, this corresponds to a step of ∆β ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 1: Particle-number restored total energy surfaces for 197Au and π = +1 (top panel) or π = −1 (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Black lines are separated by 1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The minimum for positive (negative) parity, indicated by a silver star, is located at a deformation of β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='12 and γ = 38◦ (β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='12 and γ = 19◦) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='2 Structure of 197Au 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='1 Energy surfaces The first step in our approach is the generation of a set of one-quasi-particle states that will be used as reference states in the final configuration mixing calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' To generate and select the reference states, we follow the strategy detailed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We briefly recall here that this implies: i) the self-consistent blocking of four different one-quasi-particle states at each point of the deformation mesh, ii) the projection onto good particle numbers and good angular momentum of all (converged) one-quasi-particle states, and iii) the selection of the ones having a projected energy lower than a given threshold above the projected minimum of same parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In this work, we use a threshold of 5 MeV for both positive and negative parity states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' But before discussing the final results obtained after con- figuration mixing, let us first analyze the intermediate steps in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Figure 1 displays the particle- number restored (PNR) total energy surface for the positive and negative parity states of 197Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' As can be seen, the two energy surfaces exhibit a γ-soft topography with a slightly deformed minimum2 located at β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Also, we notice that the surface for positive parity is softer at small deformation than the surface for negative parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Finally, the minimum for positive parity states is approximately 200 keV lower than the one for negative parity states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Performing the full symmetry restoration, we display in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 2 the angular-momentum and particle-number restored (AMPNR) total energy surfaces for the lowest Jπ = 1/2+, 3/2+ and 11/2− projected states, which are the three values of Jπ giving the lowest projected energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' A first remark is that the energy surfaces are much more rigid with now a well pronounced triaxial minimum with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Compared to the PNR case, the minima of the AMPNR surfaces gain rouhgly 5 MeV in binding energy and the absolute minimum is obtained for Jπ = 3/2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' It is also worth mentioning that the one- quasi-particle state giving the lowest projected state is obtained by blocking a quasi-particle that is dominated by a single-particle state originating from the spherical 2d3/2 shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The latter observations are consistent with the experimental spin-parity assignment 3/2+ 1 for the ground state of 197Au as well as its naive single-particle interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' However, we notice that the minimum for the Jπ = 3/2+ surface is located at a deformation with an angle γ = 24◦, which seems to be at variance with the oblate-like shape expected from simple argu- ments as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Nevertheless, it is important to remark that the configuration mixing may change this picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In addition, we displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 2 only the surface for the lowest Jπ = 3/2+ projected states, but given the fact that we explore triaxial deformations, all the reference states with a non-zero average value of γ will generate after angular-momentum restoration two projected states with Jπ = 3/2+ that will enter the configuration mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Ultimately, given the fact that the AMPNR is only an intermediate step in our approach, it is neither possible nor desirable to definitively charac- terize the structure of the final correlated state at this level of approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Additionally, we note that the angular-momentum pro- jection does not shift the energy minimum towards larger values of β compared to the plain PNR case, which is 2Note that all the extrema discussed in this article are com- puted from an interpolation based on the results obtained at the points on the discretized deformation mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 4 Jπ = 1/2+ Jπ = 3/2+ Jπ = 11/2− Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 2: Angular-momentum and particle-number re- stored total energy surface for 197Au and for the lowest Jπ = 1/2+ (top panel), the lowest Jπ = 3/2+ (mid- dle panel) and lowest Jπ = 11/2− (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Black lines are separated by 1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The minima for Jπ = 1/2+, 3/2+ and 11/2−, indicated by silver stars, are located at deformations of β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='13 and γ = 39◦, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='13 and γ = 24◦ and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='13 and γ = 22◦, respectively contrary to what is often observed in MR-EDF calcula- tions [30,37–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 3: Low-energy spectrum for 197Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Experimental data are taken from [46], which are based on the evalu- ation [3] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='2 Low-energy spectroscopy Finally, we perform the full configuration mixing of sym- metry projected reference states considering for positive (negative) parity a set containing 24 (19) one-quasi- particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 3 we compare the theoretical results to the available experimental data for the low- lying states up to 1 MeV of excitation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' First of all, we remark that the theory is able to reproduce the spin-parity assignment for the ground state (3/2+ 1 ) as well as for the the first (1/2+ 1 ), second (3/2+ 2 ) and third (5/2+ 1 ) excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' As in experimental data, the theory predicts a staggering between the two fomer and two latter states but the relative spacing between the two pairs of levels, as well as the spacing between the levels within a pair, are too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The 5/2+ 2 and 7/2+ 1 states also appear in our calculations but at too high excitation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The low-lying spectrum of positive parity states of 197Au has been interpreted with De-Shalit’s core-excitation model of odd-mass nuclei [47], within which an odd- even nucleus is treated as a single nucleon coupled to an even-even core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Whenever the excitation of the core is energetically favored compared to the promotion of the single nucleon to a higher orbital, in this model the lowest lying excited states of the odd-even nucleus can be interpreted as the single-particle configuration of the 5 ground state coupled in different ways to the lowest excitation of the even-even core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' When applied to 197Au [6,48–51], the ground state of the nucleus is constructed as a proton 2d3/2 particle (hole) coupled to a 196Pt (198Hg) core with Jπ = 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='3 Then, the weak coupling of the same 2d3/2 particle, or hole, to the Jπ = 2+ excited state of the (appropriate) core generates a quartet of state with Jπ = 1/2+, 3/2+, 5/2+, 7/2+ whose energy centroid Ec = {� J(2J + 1)E(Jπ)} / {� J(2J + 1)} is equal to the energy E(2+) of the excited core [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Using the experimental excitation energies, we obtain Ec = 364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='2 keV that is close to the values of E(2+) = 355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='7 keV and for 411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='8 keV for 196Pt and 198Hg, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Computing the energy centroid within our approach, we obtain the value Ec = 630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='7 keV that is obviously too large compared to the experimental one but should be compared the theoretical values of E(2+) for the neighboring even-even nuclei calculated within the same theoretical framework, which is technically possible but falls outside the scope of the present article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The MR-EDF theory also correctly predicts the 11/2− 1 state to be lowest state of negative parity but with an excitation energy of 792 keV, about 400 keV too high compared to the experimental value of 409 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This is especially surprising given that the energy difference between the AMPNR minima for Jπ = 3/2+ and 11/2− has the correct order of magnitude as can be seen Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' As a matter of fact, the minimum for Jπ = 11/2− has roughly the same energy as the one for Jπ = 1/2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' What happens is that during the configuration mixing, the 11/2− 1 state does not gain nearly as much correla- tion energy as the positive parity states and, therefore, ends up at a too high excitation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' It is not en- tirely clear why the mixing is less important in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' It might be due to the deficiency of the effective interaction but we can not exclude the possibility that other factors may play a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' For example, we had to use more restrictive values for the cutoffs before and after K-mixing to remove states not well represented on our Cartesian mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Therefore, it is possible that some important components or non-diagonal matrix elements suffer from numerical inaccuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Another possibility is that our selection strategy for the one-quasi-particle to self-consistent block at the mean-field level misses some configurations with negative parity relevant in the subsequent shape mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In general, the theoretical spectrum is too spread in energy, which is an often-encountered deficiency of MR- EDF calculations based on reference states generated by 3We mention in passing that some authors argue that using a 198Hg core provides a better global description of experimental data [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Quantity Experiment Theory E(3/2+ 1 ) 1559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='384 1556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='044 rrms(3/2+ 1 ) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='4371(38) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='389 µ(1/2+ 1 ) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='416(3) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='01 µ(3/2+ 1 ) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='1452(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='38 µ(5/2+ 1 ) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='74(6) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='15 µ(5/2+ 2 ) +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='0(5) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='14 µ(7/2+ 1 ) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='84(7) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='51 µ(9/2+ 1 ) +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='5(5) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='81 µ(11/2− 1 ) (+)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='96(9) +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='87 Qs(3/2+ 1 ) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='547(16) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='65 Qs(11/2− 1 ) +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='68(5) +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='05 Table 1: Spectroscopic quantities for the low-lying states of 197Au: total energy E (MeV), root-mean-square (rms) charge radius rrms (fm), magnetic dipole moments µ (µN), and spectroscopic quadrupole moments Qs (eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Experimental data are taken from [11,12,55–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The experimental error on the binding energy is much smaller than the rounded value given here a variation of the total energy without consideration for the angular momentum of the trial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Indeed, such a variation tend to energetically favor the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This deficiency can be in principle corrected by adding a constraint on the average angular momentum of the trial states during the minimization and using the value of the constraint as an additional generator coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Unfortunately, such calculations are computationally expensive and very few practical applications exist [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In Table 1, we report spectroscopic quantities for some of the low-lying states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' First, we see that the calcu- lations reproduce fairly well the binding energy and root-mean-square charge radius of the ground state, with a relative accuracy below 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The spectroscopic quadrupole moments for the 3/2+ 1 and 11/2− 1 states are also reasonably well described in spite of being slightly too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' While we indicate in Table 1 the value for spectroscopic quadrupole moment of the ground state, Qs(3/2+ 1 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='547(16) eb, currently taken as the accepted value in the compilation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' [58], and which was determined using muonic hyperfine measur- ments [51], we remark that other values appear in the literature that are slightly larger, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='60 eb and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='64 eb in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' [59] and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='59 eb in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' [60], and in better agreement with the value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='65 eb obtained in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Concerning the magnetic moments, they are, overall, poorly described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The values of most of them are sig- nificantly underestimated in our calculations and the 6 Transition Type Experiment Theory 1/2+ 1 → 3/2+ 1 E2 35(3) 45 M1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='019 3/2+ 2 → 1/2+ 1 E2 18(3) 6 M1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='089(9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='048 3/2+ 3 → 1/2+ 1 E2 9 M1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='34 3/2+ 2 → 3/2+ 1 E2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='5(19) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='4 M1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='002 3/2+ 3 → 3/2+ 1 E2 4 M1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='02 5/2+ 1 → 1/2+ 1 E2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='4(17) 12 5/2+ 1 → 3/2+ 1 E2 26(6) 30 M1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='034(4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='065 5/2+ 2 → 1/2+ 1 E2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='6(23) 8 5/2+ 2 → 3/2+ 1 E2 7(6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='4 M1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='083(10) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='001 7/2+ 1 → 5/2+ 1 E2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='18(7) 1 M1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='012(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='106 7/2+ 1 → 3/2+ 1 E2 33(3) 38 7/2+ 1 → 3/2+ 2 E2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='8(20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='3 7/2+ 1 → 3/2+ 3 E2 3 7/2+ 2 → 3/2+ 2 E2 6(4) 22 7/2+ 2 → 3/2+ 3 E2 2 7/2+ 2 → 5/2+ 1 E2 21(6) 13 M1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='175(23) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='010 9/2+ 1 → 7/2+ 1 E2 10(7) 10 M1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='028(10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='047 9/2+ 1 → 5/2+ 1 E2 41(5) 43 Table 2: Reduced transition probabilities among the low-lying state of 197Au given in Weisskopf units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Ex- perimental data are taken from [46], which are based on the evaluation [3] moment of the ground state has even the wrong sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Surprisingly, the best (relative) agreement with exper- imental data is obtained for the magnetic moment of the 11/2− 1 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' A similar mediocre description of the magnetic moments was already observed in our study of the 128,129,130Xe nuclei and we refer to this article [34] for a discussion of the large spectrum of possible reasons for this problem that is faced by the vast majority of EDF calculations of magnetic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In Table 2, we compare the theoretical values for the reduced transition probabilities B(E2) and B(M1) to available experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Concerning the E2 transi- tions, the theory gives reasonable estimates for most of the decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In particular, all of the strong transitions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 1/2+ 1 → 3/2+ 1 , 5/2+ 1 → 3/2+ 1 , 7/2+ 1 → 3/2+ 1 and 9/2+ 1 → 5/2+ 1 , are well described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' More generally, the hierarchy between the transitions seems to be respected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' strong (weak) experimental transitions tend to be strong (weak) in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' One notable excep- tion are the transitions towards/from the 3/2+ 2 state that are largely underestimated in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' A possible interpretation is that the 3/2+ 2 and 3/2+ 3 states are inverted in our calculation compared to the experi- mental spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Indeed, in Table 2 we also report the calculated transitions towards/from the 3/2+ 3 state that are in better agreement with the experimental data for the transitions towards/from 3/2+ 2 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In the limit case of the core-excitation model discussed above, the reduced transition probabilities from the states of quar- tet 1/2+ 1 , 3/2+ 2 , 5/2+ 1 , 7/2+ 1 to the 3/2+ 1 ground state are supposed to be equal among each other and with the B(E2 : 2+ 1 → 0+ 1 ) of the even-even core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Obviously, these equalities are not verified exactly for experimen- tal data but the values remain somewhat close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='4 In particular, within the same model, the electromagnetic transition probabilities are very sensitive to the mixing of the Jπ = 3/2+ intrinsic states [48], a problem that might also be present in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Concerning the M1 transitions, the model performs poorly and most of the probabilities are either widely underestimated or widely overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' These lacking results are consistent with the observation made above on the magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Again, this characteristic is a deficiency found in many nuclear EDF calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' While the projection techniques used here are crucial for the reliable and unambiguous comparison of calculated and experimental data for magnetic properties, they do by themselves not lead to a satisfying description of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We refer again to [34] for further discussion of this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='3 Collective wave functions We now turn our attention towards the analysis of the collective wave functions gJπ σ (β, γ) of the correlated states as defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We recall here that the squared collective wave functions (scwf) is a quantity that can be used to gauge the importance of a given deformation in the correlated wave function obtained in the final step of the MR-EDF calculations, with the caveat that, strictly speaking, it cannot be interpreted as a probability distribution due to the non-orthogonality of the reference states in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 4, we display the scwf for several low-lying states of 197Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Interestingly, in all cases, the distribution of the scwf squared is dominated by triaxial shapes, 4We mention that the B(E2 : 2+ 1 → 0+ 1 ) values are 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='6(20) and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='8(4) W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' for 196Pt and 198Hg, respectively [46,61,62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 7 Jπσ = 1/2+1 Jπσ = 3/2+1 Jπσ = 3/2+2 Jπσ = 3/2+3 Jπσ = 5/2+1 Jπσ = 5/2+2 Jπσ = 7/2+1 Jπσ = 7/2+2 Jπσ = 9/2+1 Jπσ = 11/2− 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 4: Collective wave function squared for several low-lying states of 197Au with different values of Jπ σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Black lines are separated by 10% of the (respective) maximum value indicated by a silver star with a sharply peaked maximum that has a quadrupole deformation of β ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' But depending on the value of Jπ σ , the maximum is either located at angle γ ≈ 40◦ or γ ≈ 20◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In particular, we remark that the scwf of the 3/2+ 1 and 3/2+ 2 states exhibit different behaviour with the former being located closer to the oblate axis whereas the latter favours the prolate side of the sextant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Also, the scwf of the 3/2+ 3 state is very similar to the one of the 3/2+ 1 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This is interesting for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' First, it is contrary to what could have been expected looking at the AMPNR energy surface for Jπ = 3/2+ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Second, this is consistent with the oblate-like behavior expected in an independent-particle model for a nucleus close to the Z = 82 and N = 126 shell closures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Still, it is important to stress that non-axial deformations carry a substantial percentage of the scwf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Looking more closely at the scwfs of the positive parity states, we can arrange them into three groups of similar appearance: a) the states 1/2+ 1 , 3/2+ 1 , 3/2+ 3 and 7/2+ 1 that have a scwf mostly located in the range 30◦ ≤ γ ≤ 60◦ b) the states 3/2+ 2 , 5/2+ 2 and 7/2+ 2 that have a scwf mostly located in the range 0◦ ≤ γ ≤ 30◦ and c) the states 5/2+ 1 and 9/2+ 1 whose scwf are more evenly distributed as a function of γ and seem a combination of the cases a) and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This is in good agreement with the data for the reduced transition probabilities given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Indeed, the transitions between the states of a given group have very large B(E2) values whereas the transitions between the states belonging to a different group are less likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This is not a perfect rule, however, because the 5/2+ 1 state has also an strong transition towards the 3/2+ 1 ground state but not towards the 1/2+ 1 excited state even if the scwfs of the two latter states have similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' To come back to the core-excitation model analysis, the fact that the scwf of the 1/2+ 1 , 3/2+ 3 ,5 5/2+ 1 and 7/2+ 1 excited states have a large overlap with 5Provided that we interpret the 3/2+ 2 and 3/2+ 3 states as being inverted in our calculations compared to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 8 the scwf of the 3/2+ 1 ground state is consistent with the interpretation of the quartet of positive parity state as being weak coupling of the same single-particle state to a collective even-even core with an angular momentum of either Jπ = 0+ or 2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' As a last comment, we remark that the scwf of the 11/2− 1 state has a narrower distribution than the other ones displayed, which is consistent with the fact that this state does not mix as much when diagonalizing the Hamiltonian within the space spanned by the symmetry- projected reference states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='4 Average deformation Finally, following the strategy presented in our previous article on xenon isotopes [34], we use the scwf to compute deformation parameters for the 3/2+ 1 ground state of 197Au and obtain: an average elongation of ¯β(3/2+ 1 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='13, with a standard deviation of ∆β(3/2+ 1 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='03, and an average angle of ¯γ(3/2+ 1 ) = 40◦, with a standard deviation of ∆γ(3/2+ 1 ) = 15◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This average deformation is consistent with the distribution displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 4 as the maximum is located at a deformation of β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='14 and γ = 41◦ but the distribution extends towards smaller values of β and is more or less equally distributed with respect to the γ = 40◦ axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Within the rigid rotor model, it is also possible to com- pute a deformation βr for the 0+ 1 ground state of an even-even nucleus using the experimental B(E2) values, for more details see for example Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' [34,63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Comput- ing βr for the even-even nuclei adjacent to 197Au one obtains the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='13 for 196Pt and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='11 198Hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Our average deformation ¯β(3/2+ 1 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='12 fits nicely between these two values, although we have to mention that the definitions of the two elongations are model depen- dent such that this excellent agreement may be partly accidental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The results of axially-symmetric EDF calculations based on the Gogny D1S parametrization [15,16] reported in the AMEDEE database [17] indicate a sharp minimum at a deformation of about β ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='12 for 197Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This is perfectly consistent with our estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The AMEDEE database also reports average deformations obtained from large-scale five-dimensional collective Hamiltonian (5DCH) calculations of even-even nuclei throughout the nuclear chart [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We recall here that the 5DCH can be derived as an approximation to the full GCM performed here [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' While their definition for the average deforma- tion differs from ours, we mention that for 196Pt (198Hg), they obtain an average elongation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='135 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='110), with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='032 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='030), and average angle of 32◦ (31◦), with a standard deviation of 12◦ (12◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' If the values for the elongation are consistent with our result, the average angles differ slightly with the 5DCH result indicating a deformation right at the center of the triaxial plane, although the fluctuations are large enough such that the results are compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 3 Heavy-ion collisions As previously mentioned, knowing the structure of 197Au is of particular relevance in the context of high-energy nuclear experiments, as gold is the primary species col- lided at the BNL RHIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This section analyzes the con- sequences of our results for model simulations of ultra- relativistic 197Au+197Au collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='1 Woods-Saxon parameterization of the ground state Traditionally, simulations of high-energy nuclear col- lisions take as input from nuclear structure a point- nucleon density which is used to sample nucleon coordi- nates and define an interaction region between two ions on a collision-by-collision basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='6 The standard choice for the nucleon density is that of a deformed Woods-Saxon (WS) profile: ρ(r, θ, φ) = ρ0 1 + e[r−R(θ,φ)]/a , (1) where r, θ, φ are the usual spherical coordinates, ρ0 is the saturation density, a is the surface diffuseness and R(θ, φ) is the nuclear radius parameterized as R(θ, φ) = R0 � 1 + βWS 2 � cos(γWS)Y20(θ, φ) (2) + √ 2 sin(γWS)Re � Y22(θ, φ) �� + βWS 4 Y40(θ, φ) � , where the spherical harmonics Ylm(θ, φ) are in complex form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Note that the shape parameters βWS 2 , γWS and βWS 4 represent surface deformations that differ from the volume deformation reported in the analysis of the previous sections [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We consider now the intrinsic shape of 197Au computed from a single Hartree-Fock-Bogoliubov (HFB) calcula- tion with the SLyMR1 interaction in which the expecta- tion value of the quadrupole operators are constrained 6More sophisticated calculations based on nuclear configura- tions obtained from ab initio nuclear theory have also been recently performed [65–68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' For the moment, they are limited to the description of collisions of 16O ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 9 Parameter Proton Neutron Nucleon ρ0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='157 R0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='44 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='65 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='56 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='48 βWS 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='135 γWS 43◦ 43◦ 43◦ βWS 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='023 Table 3: Parameters for the point-proton, point-neutron and point-nucleon densities defined as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' (1) and fitted to reproduce the one-body densities of a quasi- particle state constrained to have, on average, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='13 and γ = 40◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' see the body of the text for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The parameters R0 and a are given in units of fm, whereas ρ0 is given in units of fm−3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 5: Schematic illustration of the shape of 197Au based on the surface parametrization of the matter density of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' (2), and using the parameters reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 3 such that the one-body density of the trial one-quasi- particle state7 verifies, on average, β = ¯β(3/2+ 1 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='13 and γ = ¯γ(3/2+ 1 ) = 40◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='8 We fit the resulting one- body nucleon density with the Woods-Saxon profile given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The fit parameters are reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We obtain, thus, a new microscopically mo- tivated parametrization for the Woods-Saxon profile representing the nucleon density of the ground state of 197Au which can be employed in simulations of high- energy collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This profile corresponds to a triaxial ellipsoid with radii 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='02 fm, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='68 fm, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='97 fm, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 7The trial one-quasi-particle state is built by blocking a single- particle state originating from the spherical 2d3/2 shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 8All other non-vanishing multipole moments authorized by the symmetries of our calculations are let free to adopt a value that minimizes the total energy of the trial quasi-particle state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' For completeness, we evaluate as well the neutron skin of the intrinsic shape, as defined by the difference of rms radii, ∆rnp = ⟨r2⟩1/2 n −⟨r2⟩1/2 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' For the density returned by the constrained HFB calculation, we find ∆rnp[HFB(¯β, ¯γ)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='17 fm, (3) which is in perfect agreement with the result obtained from the full MREDF calculation ∆rnp[MREDF] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='17 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' (4) On the other hand, the fitted Woods-Saxon profile gives a neutron skin ∆rnp[WS fit] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='19 fm, (5) meaning that, even for a large nucleus such as 197Au, the Woods-Saxon parametrization does not fully capture skin differences of order 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='1 fm between neutrons and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We note that both the above estimates agree with a recent measurement of the STAR collaboration obtained via diffractive photo-production of ρ0 mesons in ultra-peripheral 197Au+197Au collisions [70], ∆rnp[STAR] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='03 (stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=') ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='08 (syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=') fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' (6) We note, in addition, that the half-width radius obtained for 197Au by the STAR collaboration, R0[STAR] = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='06 fm, is fully consistent with that exhibited by our nucleon density, R0[WS fit] = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='56 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This suggests that the density of gluons relevant for scattering at these beam energies is in fact very close to the rest- frame point-nucleon density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This potentially adds to the circumstantial evidence of a small nucleon width in high-energy collisions mediated by gluons [71–75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We discuss now the observational consequences of our newly-derived nucleon density for relativistic 197Au+197Au collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Model calculations of such processes (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' [76] for a state-of-the-art Bayesian analysis) have so far employed the charge density of the nucleus, as inferred from low-energy electron-nucleus scattering ex- periments [77], as a proxy for the nucleon density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The corresponding radial profiles are R0 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='38 fm, and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='53 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Nuclear quadrupole deformation has been instead included by simply implementing βWS 2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='13, as reported by finite-range liquid drop model evalua- tions [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In terms of radial profiles, there are, thus, minor differences between the WS parametrization that we show in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 3 and that implemented in the lit- erature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We only note a reduction by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='05 fm in the diffuseness parameter, a, which is due to the inclusion of the neutron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This will have a mild, though visible impact on the initial eccentricities, εn, of the sys- tem [79–81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' A new feature of our calculation is instead the fact that 197Au is not fully oblate, but presents γWS = 43◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We investigate now the impact of such a feature on high-energy collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 10 cm 100% x1013 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='02 fm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='68 fm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='97 fm10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='2 Impact of the triaxiality In the context of multi-particle correlation measure- ments in the soft sector of high-energy nuclear colli- sions, the strongest sensitivity to the triaxial structure of the colliding nuclei is carried by the mean momentum- elliptic flow correlation [82–84], ρ2 ≡ ρ(⟨pt⟩, v2 2) = ⟨⟨pt⟩v2 2⟩ − ⟨⟨pt⟩⟩⟨v2 2⟩ σ(⟨pt⟩)σ(⟨v2 2⟩) , (7) where outer brackets denote a statistical average over events, and σ(o) is the standard deviation of observable o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This quantity can be evaluated in the final states as a three-particle correlation [85], and it measures the strength of the statistical correlation between the charged-particle average transverse momentum, ⟨pt⟩, and the charged-particle elliptic flow, v2, at a given collision multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' To assess the impact of γWS = 43◦ on the ρ2 correlator of 197Au+197Au collisions, we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' [30] and provide an estimate of the measured ρ2 from high-statistics simu- lations of the initial condition of these processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' For the details of such simulations, we refer to the exhaustive de- scriptions given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Briefly, we assume that the distribution of final-state multiplicities is proportional to the distribution of initial-state entropy, S, which we calculate event-to-event following the original TRENTo parametrization [86] (s(x, τ0) ∝ √TATB, S = � d2x s) with a nucleon size w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='5 fm, and a fluctuation pa- rameter, k, tuned to reproduce measured multiplicity histograms in 208Pb+208Pb collisions at CERN LHC energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We consider that i) the mean transverse momen- tum is, at a given entropy, proportional to the initial E/S, where E is the total energy of the system [87,88], obtained upon application of the equation of state of high-temperature QCD (e(x) ∝ s(x)4/3, E = � d2x e), and ii) that the elliptic flow is proportional to the initial eccentricity of the system, ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The Pearson correlation coefficient of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' (7) can then be estimated by replac- ing v2 2 and ⟨pt⟩ with, respectively, ε2 2 and E/S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Note that the resulting estimator should not be compared directly to the experimental measurements, as it misses effects related to the cuts in transverse momentum, pt, implemented in the experimental analysis, which have been shown to be sizable for the magnitude of this observable [31,89,90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' That said, it is the initial-state estimator that carries the dependence on the deforma- tion parameters, such that the relative impact of the value of γWS on the final-state result can be assessed from it [30,91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We perform 20 × 106 minimum bias simulations of 197Au+197Au collisions for three structure scenarios, 300 350 400 450 500 550 600 Nrec ch (|η| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='25 ρ � ⟨pt⟩, v2 2 � ← uncertainty on STAR data at Nrec ch ≈ 550 TRENTo, 200 GeV Au+Au oblate gold (βWS 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='135, γWS = 60◦) triaxial gold (βWS 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='135, γWS = 43◦) prolate gold (βWS 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='135, γWS = 0) 16 9 3 1 centrality (%) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 6: Initial-state estimates of ρ(⟨pt⟩, v2 2) in 200 GeV 197Au+197Au collisions for prolate ions (dot-dashed line), oblate ions (dotted line) and triaxial ions (dashed line) presenting γWS = 43◦, as a function of the number of reconstructed charged tracks in the STAR detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Shaded bands (of the same width as the lines) are statis- tical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The figure reports as well the total uncertainty on preliminary STAR measurements for this observable at high multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' namely, we set βWS 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='135, and consider γWS = 0◦, 43◦, and 60◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='9 Rescaling the TRENTo entropy to match the observed mutliplicity of reconstructed charged tracks in the STAR detector, N rec ch , at midrapidity (|η| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='5), our results for ρ2 are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Qualitatively, the impact of γWS follows the generic parametric expec- tation ρ2 ∝ c0 − c1(βWS 2 )3 cos(3γWS), where c0 and c1 are positive coefficients [30,91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We conclude that a 17◦ deviation from oblateness in 197Au leads to a correction of order 10-15% to ρ2 for collisions in the 0-2% centrality range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We reiterate that, while our results for the magni- tude of the Pearson coefficient should not be compared directly to data, we expect the correction induced by the triaxiality, relative to the oblate scenario, to be ro- bustly captured by our initial-state evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 6 we report as well the size of the experimental error on preliminary ρ2 data at high multiplicity from the STAR collaboration [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The error bar turns out to be signifi- cantly smaller than the splitting that we find between the triaxial scenario (red dashed line) and the oblate scenario (dotted blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Therefore, according to our results the impact of the triaxiality has been already iso- lated in the preliminary data, and it will be possible to 9We safely neglect the effect of the very small hexadecapolarity of the nucleus, βWS 4 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='023, in these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 11 quantify it in the future via high-precision hydrodynamic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' We stress, though, that the most effective way to access the value of γWS is by studying the ρ2 correlator of 197Au+197Au collisions normalized with that of 238U+238U collisions, as done in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' [30, 31] to extract such an information in the comparisons of 129Xe+129Xe and 208Pb+208Pb collisions, which allows one to fully cancel theoretical and experimental system- atical uncertainties and isolate transparent information about the nuclear structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The current mismatch be- tween hydrodynamic results and experimental data for 238U+238U collisions [93] prevents us, for the moment, from performing such an analysis, which will be thus reported in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 4 Conclusions In the present article, we first reported on new re- sults on the low-energy structure of the heavy odd- mass nucleus 197Au obtained by performing state-of- the-art MR-EDF calculations that include the mixing of angular-momentum and particle-number projected Bogoliubov quasi-particle states with different average triaxial shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' All the calculations were realized using the parametrization SLyMR1 of a Skyrme-type pseudo- potential [44,94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Although odd-mass nuclei represent half of the existing nuclei in the nuclear chart, their calculations within the full-fledged MR-EDF framework are still scarce, ex- ceptions being [34,40,41,95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In this work, to generate reference states adapted to the modeling of odd-mass nuclei, we performed self-consistent blocking of Bogoli- ubov one-quasi-particle states and considered exactly all the time-odd terms of the functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The results obtained on the low-energy spectroscopy of 197Au are reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The spin-parity assignments for the 3/2+ 1 ground state and for the first few excited states are correct even if the levels are too spread out, a well-known deficiency of usual MR-EDF calculations that can be corrected by adding a supplemental con- straint on the average angular momentum of the trial wave functions when generating the set of reference states to be projected and mixed [53,54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' The binding energy, root-mean-square charge radius and spectro- scopic quadrupole moment of the of the ground state are also well reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' By contrast, the calculations fail to reproduce the known magnetic moments for the ground and excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Concerning the electromag- netic transitions, the values for the reduced transition probabilities B(E2) are, overall, well described whereas the values for the B(M1) are off, sometimes by more than one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Starting from the collective wave function of the ground state, we computed average triaxial deformation param- eters ¯β(3/2+ 1 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='13 and ¯γ(3/2+ 1 ) = 40◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Following the the strategy of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' [30], we then fitted the parameters of a deformed Woods-Saxon density profile, to obtain a new state-of-the-art microscopically-motivated input for the simulation of high-energy 197Au+197Au colli- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' In terms of radial profile parameters, our result corrects to some extent the widely- and incorrectly- employed charge-density parametrization, which has in particular a too large skin thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' For future precision phenomenological studies of 197Au+197Au collisions, es- pecially in view of the upcoming sPHENIX program at the BNL RHIC, it will be crucial to implement real- istic properties of the point-nucleon density in Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This includes as well implementing an appropriate triaxiality, of order 45◦, for gold ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Our estimates indicate that this magnitude of the tri- axiality does impact the final state in a significant way, and we expect future theoretical work to be able to cleanly isolate such a contribution from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' As an outlook, we emphasize that measurements of the third centered moment (skewness) of the distribution of ⟨pt⟩ [96] provide additional and independent information about γWS [91], and can be used in conjunction with hydrodynamic simulations to further test our prediction for this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' Acknowledgements We thank Chunjian Zhang for help with the entropy-to-multiplicity conversion used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 6, and Wouter Ryssens for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' This project has re- ceived funding from the European Union’s Horizon 2020 re- search and innovation programme under the Marie Sk�lodowska- Curie grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 839847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' acknowledges sup- port by the Agence Nationale de la Recherche, France, un- der grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 19-CE31-0015-01 (NEWFUN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' is funded by the Deutsche Forschungsgemeinschaft (DFG, German Re- search Foundation) under Germany’s Excellence Strategy EXC2181/1-390900948 (the Heidelberg STRUCTURES Ex- cellence Cluster), within the Collaborative Research 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content='103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} +page_content=' 024910' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfggHn/content/2301.02420v1.pdf'} diff --git a/E9E2T4oBgHgl3EQfSweS/content/tmp_files/2301.03796v1.pdf.txt b/E9E2T4oBgHgl3EQfSweS/content/tmp_files/2301.03796v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..64166d1b1ab0651d90cb799a4f23a8a339889290 --- /dev/null +++ b/E9E2T4oBgHgl3EQfSweS/content/tmp_files/2301.03796v1.pdf.txt @@ -0,0 +1,1067 @@ +Assessing the applicability of common performance +metrics for real-world infrared small-target detection +Saed Moradi1, Alireza Memarmoghadam1, Payman Moallem∗1, and Mohamad Farzan +Sabahi1 +1Department of Electrical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, Iran +Abstract +Infrared small target detection (IRSTD) is a challenging task in computer vision. During the last +two decades, researchers’ efforts are devoted to improving detection ability of IRSTDs. Despite the huge +improvement in designing new algorithms, lack of extensive investigation of the evaluation metrics are +evident. Therefore, in this paper, a systematic approach is utilized to: First, investigate the evaluation +ability of current metrics; Second, propose new evaluation metrics to address shortcoming of common +metrics. To this end, after carefully reviewing the problem, the required conditions to have a successful de- +tection are analyzed. Then, the shortcomings of current evaluation metrics which include pre-thresholding +as well as post-thresholding metrics are determined. Based on the requirements of real-world systems, +new metrics are proposed. Finally, the proposed metrics are used to compare and evaluate four well- +known small infrared target detection algorithms. The results show that new metrics are consistent with +qualitative results. +Keywords: Infrared small target detection; thresholding; pre-thresholding metrics; post-thresholding +metrics +1 +Introduction +Nowadays, infrared (IR) imaging has a wide range of application from medical [1, 2] and industrial diagnosis +[3] to defense [4] and remote sensing [5]. Generally, processing IR images is a challenging task [6] due to the +specifications of IR imaging. Among all the aforementioned applications, IR small target detection (IRSTD) +is a highly challenging research field because: +• Since the IR small targets are far from the imaging device, the target has low local contrast and appears +as a dim spot in the image plane [7]. +• The small target in IR images typically occupies handful of pixels [8]. Thus, the region of interest +(ROI) does not represent distinguished features. +• The edges of the small target are blurred due to atmospheric thermal fields [9]. Therefore, there are +not a clear boundary between background area and target pixels. +The block diagram of a typical IRSTD pipeline is illustrated in Fig. 1. +As shown in the figure, the +input IR image is first process by the IRSTD algorithm to create a saliency map. Note that, while the +IRSTD algorithm may refer to the end-to-end IR image processing pipeline, here, the process of construction +of saliency map from input IR image is called IRSTD algorithm. The goal is to suppress the background +area and enhance target pixels. An ideal saliency map should eliminate the background intensities and only +preserve the target area. After saliency map reconstruction, a thresholding strategy is chosen to be applied +on the saliency map. Then, true (logical one) pixels in the resulting binary image is considered as target-like +objects. +Considering the pipeline in the Fig. 1, specific attributes are defined for images in pipeline. In IRSTD +terminology, the input image can be represented by two attributes: +∗corresponding author: p moallem@eng.ui.ac.ir +1 +arXiv:2301.03796v1 [cs.CV] 10 Jan 2023 + +Input infrared image +Saliency map +Target-like candidates +IRSTD algorithm +Thresholding +Input attributes +Pre-thresholding attributes +Post-thresholding attributes +SCRin , σb,in +SCRout , σb,out +Pfa , Pd +Figure 1: The block diagram of a typical IRSTD pipline +• σb,in which denotes the standard deviation of background pixels in input image. This parameter directly +related to the background complexity. Smaller σb,in represents smooth backgrounds, while larger σb,in +belongs to a complicated background. +• SCRin stands for signal to clutter ratio in the input image. SCR is defined as µt−µb +σb +. Where, µt, µb, +and σb are the mean value of target pixels, mean value of the local background pixels, and standard +deviation of the local background, respectively. +Same argument is valid for saliency map (The processed image by IRSTD algorithm). Thus, just like the +input attributes, σb,out and SCRout represents the background complexity and the signal to clutter ration in +the saliency map. It is clear that for a typical IRSTD pipeline: +σb,out < σb,in +and +SCRout > SCRin +(1) +According to Eq. 1, two performance metrics are defined for evaluation of IRSTD algorithms: Background +suppression factor (BSF) and signal to clutter ration gain (SCRG) which are defined as follows [10]: +BSF = σb,in +σb,out +, +SCRG = SCRout +SCRin +(2) +Based on Eq. 1 and Eq. 2, larger values for both SCRG and BSF are desired. Note that, in case of +evaluation of different IRSTD algorithms, since the input images are the same for all baseline algorithms, +SCRout and +1 +σb,out can be used as performance metrics, as well. +The IRSTD algorithms are well-studied in the literature. Mainly, these algorithms can be categorized +based on filtering method, contrast measure calculation, and data structure decomposition [11]. +The filtering based methods are divided into two sub-categories. +The first one is the spatial domain +filtering, in which, the input infrared image is processed using local kernels to enhance the target area. Max- +mean [12], max-median [12], bilateral filtering [13], morphological operators [14], two dimensional least mean +square [15] are some instances of this sub-category. The second one refers to processing in the transformation +domain. +In these techniques, the input image is transformed to a desired transformation space like as +frequency [16] and wavelet [17] domains. Then, after processing the transformed information, the inverse +transform is applied to recover true targets. +Methods based on human visual systems (HVS) which lead to local contrast-based mechanism has received +researchers’ attention during last few years. These methods outperform filter-based methods in terms of +SCRG and BSF. However, they usually have higher computational complexity compared to filter-based ones. +Generally, local contrast can be constructed in either difference or ratio forms. Difference local contrast like +as Laplacian of Gaussian (LoG) [18], difference of Gaussian (DoG) [19], improved difference of Gabor [20], +center-surround difference measure [21], and local difference adaptive measure [22]. Unlike the difference form +local measures, ratio-form local measures utilize enhancement factor which is the ration between the center +cell and surrounding ones. Local contrast measure (LCM) [23], improved local contrast measure (ILCM) [24], +2 + +relative local contrast measure (RLCM) [25], Tri-Layer local constrast method (TLLCM) [26], novel local +contrast descriptor (NLCD) [27], and weighted strengthened local contrast measure (WSLCM) [28] are the +most effective IRSTDs in the literature. There is also a combined local measure which benefits from both +difference and ratio from of local contrast measure [29]. +Data structure decomposition-based methods are also a newly introduced class of IRSTDs. Sparse and +low-rank matrices decomposition is the principal of these class of IRSTDs. Infrared patch image (IPI) model +[30], weighted infrared patch image (WIPI) model [31], non-negative infrared patch image model based +on partial sum minimization of singular values (NIPPS) [32], nonconvex rank approximation minimization +(NRAM) [33], and nonconvex optimization with an Lp norm constraint (NOLC) [34] are the recent efforts of +IR image decomposition-based approach. +The goal of all aforementioned methods is to obtain larger BSF and SCRG values . However, having larger +BSF and SCRG does not guarantee a successful detection. A high performance IRSTD algorithm should +be followed by a proper thresholding strategy to detect real targets and eliminate false responses. This is +why there are two more performance metrics after applying the thresholding operation to the saliency map. +These two metrics which demonstrate the ability of detection true targets and eliminating false responses +are called probability of detection Pd and probability of false alarms Pfa, respectively. In contrast to BSF +and SCRG which are measurable before applying the threshold (This is why we call them pre-thresholding +attributes), these two metrics are measured on binary images and therefore we call them post-thresholding +attributes (Fig. 1). +As mentioned in the previous paragraph, for a successful detection, both high performance IRSTD al- +gorithm as well as the proper thresholding strategy are required. Regardless of effectiveness of the IRSTD +algorithm, improper thresholding will leads to missing true targets and having false responses which could +be disaster for a practical system. Hence, in this paper, after investigating various thresholding strategies, +the best methods for applying threshold to the saliency map is presented. Then, current pre-thresholding as +well as the post-thresholding metrics are investigated, and some new metrics which are aligned with practical +considerations are proposed. The rest of this paper is organized as follows: in the next section, the role of +thresholding in practical systems is deeply investigated. Then, in section 3, current pre-thresholding metrics +are reviewed. After demonstrating their shortages, modified metrics are proposed for IRSTD performance +evaluation. In section 4, same process is performed for post-thresholding metrics. In section 5, The newly +proposed metrics are used for performance comparison of common IRSTD algorithms. Finally, the paper is +concluded in section 6. +2 +The onus of thresholding on the overall performance +After performing target enhancement and clutter suppression procedure (saliency map construction), the +filtered IR image should be converted to binary one using thresholding operation that can be applied in +different forms (i.e manual, automatic, local, global). Since the target detection problem only consists of two +different classes namely as target and background clutter, single-level thresholding is a satisfactory option +for this purpose. The simplest method to achieve the classification goal, is to apply a global threshold T: +g(x, y) = +� +1 +f(x, y) > T +0 +f(x, y) ≤ T +(3) +where, f(x, y) and g(x, y) stand for the saliency map and binary image,respectively. The most challenging part +of global thresholding operation is how to set an effective threshold value T. Since target detection systems +continuously scan the environment, human operator cannot be helpful to choose the optimum threshold +value. The most simplest way to do this is to choose a unique threshold value based on experiments for all +incoming image frames. However, when the dynamic range of the filtered image is not equal to the dynamic +range of the input images, the false-alarm rate or the miss-rate will increase drastically. Fig. 2 shows the +change in the dynamic range of filtered images (saliency maps) using Tophat and AAGD IRSTDs. As shown +in the figure; the output dynamic range directly depends on the applied IRSTD. Therefore, the thresholding +procedure should be performed in an automated manner. There are various automatic image thresholding +algorithms in the literature. The Otsu’s method is one of the widely used one [35]. In this method the global +threshold value is chosen in a way, to maximize inter-class variance. When both foreground (Target) and +3 + +40 +60 +80 +100 +120 +140 +160 +180 +200 +(a) +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +(b) +0 +500 +1000 +1500 +2000 +2500 +(c) +Figure 2: Variable dynamic range in saliency map. +a) Original infrared image. +b) filtering result using +TopHat algorithm [14], c) filtering result using AAGD [37] algorithm. The dynamic range of the saliency +map might be different than input infrared image depending on the applied IRSTD. +background classes include considerable number of pixels, and the image histogram is bimodal (i.e. there +is a deep valley between two peaks in the image histogram), the Otsu’s method works very well in object +segmentation problems. However, when the target area is too small compared to the background area, which +is always occurred in incoming infrared target detection problems, the segmentation result of Otsu’s method +is inaccurate Fig. 3. Another widely used automatic thresholding is presented in [36], where the followings +are performed to obtain the desired threshold value: +i) The gray image is segmented into two classes using threshold value equal to global mean of the image +(T = µG). +ii) The average values of the background and target are calculated (µB, µT ). +iii) The new threshold level is calculated +� +Tnew = 1 +2 (µT + µB) +� +. +iv) While (Tnew − Told > ϵ), steps (ii) and (iii) are recursively repeated. +When the background noise is not strong, this automatic thresholding operation shows good performance +for final target detection (Fig. 3d). However, in strong noisy scenarios, the performance of this algorithm +is degraded significantly (Fig. 4d), which in turn, increases the false responses. Moreover, when infrared +scenario does not contain any small target, these histogram-based automatic thresholding methods always +return incorrect responses in non-target areas (Fig. 5). +Statistics-based image thresholding is the most effective thresholding strategy for small target detection +which can be applied in both local and global manners. Statistics-based global and local thresholding are +expressed in Eq. 4 and Eq. 5, respectively. +T = µG + kG × σG +(4) +T(x, y) = µ(x, y) + kL × σ(x, y) +(5) +where, µG, σG, µ(x, y), σ(x, y), kG, and kL indicate global mean of the image, global standard deviation of the +image, local mean around (x, y) position, local standard deviation around (x, y) position, control parameter +of global thresholding and local thresholding, respectively. +Global thresholding is a simple operation with low computational complexity. However, in multi-target +scenarios, some targets may be missed. Local thresholding can detect all targets. Since local mean and +standard deviation should be calculated for each pixel in the gray image, the local statistics-based thresholding +has higher computational complexity compared to the global one. Generally speaking, using statistics-based +thresholding has the following advantages: +• It can work with any gray-level dynamic ranges. +• The control parameter (k) can be determined by experiments to achieve reasonable false-alarm rate. +• The last but not the least, it is very effective for scenarios with no targets. +4 + +(a) +(b) +(c) +(d) +(e) +Figure 3: The automatic thresholding results. a) Original infrared image. b) Top-hat filtering result [14]. +c) Otsu’s thresholding result (T = 0.48). d) automatic thresholding using average values of background and +target classes (T = 19). e) Manual thresholding (T = 29). +(a) +(b) +(c) +(d) +(e) +Figure 4: The automatic thresholding results. a) Original noisy infrared image. b) Top-hat filtering result. +c) Otsu’s thresholding result (T = 0.5). d) automatic thresholding using average values of background and +target classes (T = 7). e) Manual thresholding (T = 10). +3 +Pre-thresholding evaluation +A detection process is successful as long as a single pixel of target area is correctly recognized. In this case, +the exact boundary extraction of the target area is not important at all. Therefore, a proper evaluation +metric should support this argument. +Signal to clutter ratio (SCR) is one of pre-thresholding metrics which shows the target enhancement +5 + + (a) +(b) +(c) +Figure 5: Drawback of automatic thresholding in scenarios with no targets. a) original infrared image which +does not contain small target. b) the result of Top-Hat filtering. c) automatic thresholding results. +ability of an IRSTD, which is defined as: +SCR = µT − µb +σb +, +(6) +where, µT , µb, and σb denote average intensity of the target area, average intensity and standard deviation of +its local surrounding background, respectively. While this evaluation measure is generally accepted in the lit- +erature, it can not correctly reflect the target enhancement capability of an IRSTD. To better understanding, +a simple scenario is provided here (Fig. 6). +Two different saliency maps are demonstrated in Fig. 6a and Fig. 6b. Fig. 6a shows the result of applying +AAGD algorithm [37] with 9 × 9 internal window. As depicted in the figure; the target area is relatively +enhanced while there are some remaining background clutter. Compared to the Fig. 6a, the second IRSTD +which again is an AAGD Fig. 6b algorithm with 3 × 3 internal window followed by a morphological erosion +with a 3 × 3 square-shape structural element, shows better target enhancement and background suppression. +As shown in Fig. 6c and Fig. 6d, the signal amplitude for the IRSTD #2 is almost twice as the one in the +IRSTD #1, which means in higher threshold values the target will be detected corectly in the second one, +while in the first one the target will be missed. One dimensional (1D) cross-section of target area in both +saliency maps are shown in Fig. 6e and Fig. 6f, respectively. To simplify the scenario, let’s approximate 1D +cross-section of the target area with closest square signal. The result is shown in Fig. 6g. As shown in the +figure: +AT1 = AT2 +2 +, +WT1 = 3 × WT2 +(7) +where, AT1, AT2 denote the target amplitude in the output of IRSTD #1 and #2. Also, WT1, WT2 show the +target width (extension) in the output of IRSTD #1 and #2, respectively. +Based on SCR formulation (Eq. 6), and simply considering zero-mean background signal, the following +relationship can be easily derived: +SCR1 = 3 +2 × SCR2 +(8) +which implies that the target detection ability of the IRSTD #1 is 50% more than that of IRSTD #1. +However, by applying a global threshold level at Tapp, the #1 algorithms does not detect the true target +(Fig. 6). It can be clearly seen that the #2 algorithm can detect the true target at the same threshold +level. In order to address this issue when global thresholding is final choice in practical system, the SCR +formulation should be modified as: +SCRglobal = maxT −µG +σG +, +(9) +where, maxT denotes the maximum gray value of the target area. +According to Eq. 4, the maximum +acceptable control parameter is equal to newly defined SCR metric: +kGmax = SCRglobal. +(10) +There are two important points regarding the Eq. 10: +6 + +0 +50 +100 +150 +200 +250 +(a) +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +(b) +0 +50 +100 +150 +200 +250 +(c) +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +(d) +0 +5 +10 +15 +0 +50 +100 +150 +200 +250 +(e) +0 +5 +10 +15 +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +(f) +Threshold Value +Tapp +Signal +Amplitude +AT1 +WT1 +Signal +width +Output of #1 detection algorithm +Threshold Value +Tapp +Signal +Amplitude +AT2 +WT2 +Signal +width +Output of #2 detection algorithm +(g) +Figure 6: A simple scenario to demonstrate the drawback of common SCR metric. a, b) Saliency map of the +IRSTD #1 and #2, c, d) target area in the saliency map of the IRSTD #1 and #2, e, f) 1D plot of target +cross-section in c and d, g) simplified 1D representation of target area in both IRSTD #1 and #2. +1. Common SCR metric is not able to correctly reflect the target detection ability. The pre-thresholding +evaluation should be performed in a global manner on the saliency maps. +2. The thresholding operation should be consistent with pre-thresholding evaluation metrics. For instance, +in our case, the global statistics-based thresholding is the right choice. +So far, it is demonstrated that the global thresholding is the right one to be applied on the saliency map. +In the next subsection, we demonstrate the drawback of the local statistics-based thresholding. +3.1 +Drawback of common local thresholding +Now, let consider the case that local thresholding is supposed to be applied on the saliency map. According +to Fig. 6, the local mean around target region can be calculated as follows: +µ(x) = AW +n +(11) +where, A, W, x, and n stand for the target amplitude, width (spatial extension), the current index and +number of samples in local neighborhood (n > W). +7 + +2 +3 +4 +5 +6 +7 +8 +9 +1 +2 +3 +4 +kL max +n=17 +n=25 +n=33 +Figure 7: kLmax versus target spatial extension W (target width in 1D case) +The local standard deviation can be calculated as: +σ(x) = +� +� +� +� 1 +n +n +� +i=1 +(y(i) − µ(x))2 +(12) +Where y(i) and µ(x) denote the saliency map samples and local mean, respectively. Since the detection +algorithm is supposed to suppress background clutter, for the sake of simplicity, we can assume that the +saliency map samples out of the target region are equal to zero. Then: +σ(x) = A +n +�� +nW − W 2 +� +(13) +The local thresholding (Eq. 5), can be rewritten as: +T(x) = µ(x) + kL × σ(x) = AW +n ++ kLA +n +�� +nW − W 2 +� +(14) +The detection process is established correctly for the threshold values lower than target amplitude (T(x) < +A). Therefore, for a successful target detection the following condition should be met: +µ(x) + kL × σ(x) < A +⇒ AW +n ++ kLA +n +�� +nW − W 2 +� +< A +(15) +The upper bound for control parameter (kLmax) to detect the target accurately can be find as follows: +AW +n ++ kLmaxA +n +�� +nW − W 2 +� += A +⇒ kLmax = +� +n − W +W +(16) +Fig. 7 shows the upper bound of control parameter versus target width. As shown in the figure, the maximum +control parameter to detect target correctly using local thresholding decreases as the target width increases. +kLmax takes its maximum value when the target width is equal to one pixel (kLmax = √n − 1 for W = 1). +This result is quite consistent with the fact that the most effective target detection algorithm should suppress +all background region and only returns a single pixel (Target centroid). +Fig. 8 shows the local threshold value which is normalized to the target amplitude ( T +A) versus different +control parameter. It is clear that, only when the ( T +A) fraction is less than one the target can be detected +correctly. Another finding which can be derived from the figure is that the maximum control parameter +decreases as the target width increases. Therefore, unlike the global case, the effective control parameter to +extract real targets and eliminate background clutter depends on the target area in the saliency map. Also, +the reasonable rang for control parameter is narrowed when the local neighborhood is decreased (Fig. 8b). +Moreover, using Eq. 5 to extract true target from saliency map leads to many false alarms. Fig. 9 shows the +8 + +0 +0.5 +1 +1.5 +2 +2.5 +3 +Local Adjustment Parameter (KL) +0 +0.5 +1 +1.5 +W=3 +W=5 +W=7 +W=9 +The target +is missed +The target +is correctly +detected +(a) +0 +0.5 +1 +1.5 +2 +2.5 +3 +Local Adjustment Parameter (KL) +0 +0.5 +1 +1.5 +2 +W=3 +W=5 +W=7 +W=9 +The target +is missed +The target +is correctly +detected +(b) +Figure 8: Local threshold value normalized to target amplitude ( T +A) versus different control parameter. a) +n = 33, b) n = 17. +(a) +(b) +(c) +(d) +Figure 9: Shortcoming of local thresholding. a) Original input image, the target area is marked by red +ellipse, b) the result of target enhancement using multi-scale Laplacian of Gaussian (LoG) method, c) the +local thresholding applied on (b) with k = 4, c) the local thresholding applied on (b) with k = 5. +local thresholding on the saliency map of multi-scale Laplacian of Gaussian (LoG) method [18]. As shown in +the Fig. 9b, the target area is the most salient region in saliency map. However, after thresholding using local +method (Fig. 9c), there are too many false responses. The only way to limit false responses to an acceptable +range is to increase the control parameter. However, the true target is not extracted when the local threshold +is increased. Note that there are still too many false responses in Fig. 9d. +Based on the local thresholding results Fig. 9, this method (Eq. 5) is not a proper strategy to discriminate +target area from background clutter. +4 +Post-thresholding evaluation +After applying a predefined threshold to the saliency map, a binary image is obtained. In this case, the +prevalent metrics to evaluate the performance of the detection algorithms are probability of false-alarm Pfa +and detection Pd. These two metrics are defined as [38]: +Pfa = Nf +Ntot +, +Pd = Nd +Nr +(17) +9 + +10 +−10 +10 +−5 +10 +0 +0 +0.2 +0.4 +0.6 +0.8 +1 +Pfa +Pd + + +Alg. 1 +Alg. 2 +Figure 10: ROC curve for two typical detectors +(a) +(b) +(c) +Figure 11: a) original image, b) the LCM filtering result, c) the Top-Hat filtering result. +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Pfa +0.6 +0.65 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +PD +LCM +TopHat +(a) +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +T +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Pfa +LCM +TopHat +(b) +Figure 12: a) The ROC curve, b) false-alarm rate versus different threshold levels. +where Nf, Ntot, Nd, and Nr denote the number of wrongly detected pixels, the total number of pixels, the +number of pixels which are detected correctly, and the target pixels in the ground-truth, respectively. The +receiver operational characteristics (ROC) curve is constructed by considering each (Pfa, Pd) pair at different +threshold level. +Fig. 10 shows the ROC curve for two typical detectors. As shown in the figure, for a constant false-alarm +rate, the detector #1 has higher detection rate, and outperforms the algorithm #2. The ROC curve is a +satisfactory tool to evaluate the performance of different detectors. However, if the detection rate and false- +alarm rate are not defined accurately, the final ROC curve is not a reliable measure anymore. In order to +demonstrate the deficiency of the definitions of the Pd and Pfa (Eq. 17), let consider the target detection +ability of two well-known small infrared target detection algorithms; Local contrast method (LCM) [23] +and Top-hat algorithm [39]. Fig. 11 shows the detection results of these two algorithms. As shown in the +figure, the Top-hat filtering method clearly outperforms the LCM algorithm. however, the ROC curve gives +contradictory result against visual perception (Fig. 12a). Also, by constructing the curve of the false-alarm +rate versus different threshold levels (Fig. 12b), the low performance of the LCM algorithm is clearly seen. +Therefore, the former definition of the Pd (Eq. 17) is not appropriate for this crucial metric. +Another alternative definition for Pd is suggested in the literature ([25]): +Pd = ND +NR +(18) +where ND and NR are number of detected true targets, and total number of true targets. While this new +definition addresses the deficiency of the former one (Eq. 17), there are still some drawbacks regrading this +formula; The real infrared scenarios usually contain limited number of targets. To overcome this drawback, +10 + +(a) +(b) +(c) +(d) +Figure 13: a) Synthetic targets in homogeneous local background, b) low contrast targets in background +clutter edges, c) the character filter response to a and b). +(a) +(b) +(c) +(d) +Figure 14: a, c) real infrared scenario, b , d) the response of character filter [40] to a and c, respectively. +synthetic targets are usually created using Gaussian spatial distribution. However, spatial distribution-based +target detection algorithms directly benefit from synthetic data, so the final evaluation is not fair. An example +is provided here to better demonstration of this situation. The character filter [40] utilizes Gaussian spatial +distribution as a measure to distinguish between real target and background clutter. As shown in Fig. 13, +when the small target has exactly Gaussian distribution, the character filter effectively can enhance the small +targets and eliminate background clutter. However, in real infrared scenarios, which the spatial distribution +of small targets does not follow the Gaussian distribution [38], the detection results of character filter is +chaotic (Fig. 14). +According to aforementioned issues regarding the post-thresholding performance evaluation metrics, +herein, awe present a new approach capable of addressing all the shortcomings. Since in a successful de- +tection operation, at least one pixel is detected after thresholding operation, the following procedure is +introduced to obtain new post-thresholding performance measure: +i) The upper bound for control parameter (kmax) is calculated (Eq. 10). +ii) The [0 − kmax] interval is chosen as valid interval for performance evaluation. +iii) For each different control parameters, the false-alarm rate is calculated using Eq. 17. +11 + +Table 1: The baseline algorithms +Detection Algorithm +Details +Top-Hat [39] +7 × 7 structural element +LoG [18] +With [0.50, 0.60, 0.72, 0.86, 1.03, 1.24, 1.49, 1.79, 2.14, 2.57, 3.09, 3.71] scale parameters +PCM [41] +With [3 × 3, 5 × 5, 7 × 7, and 9 × 9] cell-sizes +AAGD [42] +With [3 × 3, 5 × 5, 7 × 7, and 9 × 9] cell-sizes +Table 2: The value of maximum control parameter kmax for different algorithms +the 1st test image +the 2nd test image +the 3rd test image +the 4th test image +the 5th test image +the 6th test image +AAGD +39.8553 +61.9809 +45.8804 +57.9033 +8.0117 +166.4851 +LoG +16.5312 +28.8976 +5.4743 +7.2515 +13.4517 +37.4031 +TopHat +15.8858 +22.8994 +2.5513 +3.8742 +14.4408 +26.9160 +PCM +13.7368 +67.1190 +25.5538 +14.4819 +12.4008 +87.0127 +iv) The false-alarm rate versus control parameter (Pfa – k) curve is constructed. In the next step, the [0 +– k] interval is linearly mapped to [0 – 1] range. This normalization allows us to fairly compare and +evaluate different algorithms. +After constructing (Pfa – k) curve, the following measures can be extracted: +• The maximum control parameter (kmax) is the first inferred performance evaluation metric. The larger +kmax, the higher detection ability. +• The false-alarm rate at kmax, which is called Pfa,min here, is the second evaluation metric. It is obvious +that the false-alarm rate of the system can not be less than Pfa,min while the true target is detected. +After normalizing [0 – k] interval to [0 – 1] range, the false alarm rate of the detection algorithms can +be plotted in single figure. Then, the algorithm with satisfying detection performance can be chosen for the +practical application. +5 +Detection ability evaluation using new metrics +In order to evaluate the detection ability using the proposed metrics, four well-known small infrared target +detection algorithms are chosen to conduct the experiments. Tab. 1 reports the baseline algorithms and their +implementation details. The pre-thresholding enhancement results of each algorithm are depicted in Fig. 15. +Visually speaking, the AAGD algorithm has better performance in background suppression (the background +region is mapped to zero value). However, the most part of gray area in PCM output have zero values (Since +there are also negative values in the saliency map, the zero values are depicted by gray color instead of black +one). LoG and TopHat filters are sensitive to noise and sharp edges, therefore, there are too many false +responses in their saliency maps. +The results of evaluation using new metrics are reported in Tab. 2 and Tab. 3. As reported in Tab. 2, +AAGD and PCM algorithms have better enhancement for target area. However, by taking the false-alarms +into account, the PCM algorithm shows better clutter rejection ability. +Finally, Fig. 16 shows the normalized (Pfa – k) curve to investigate the detection performance charac- +teristics of different baseline algorithms, and fairly compare them. As shown in the figure, the PCM and +AAGD algorithm has overall superiority compared to LoG and TopHat algorithms. The AAGD algorithm +Table 3: The value of minimum probability of false alarm Pfa,min for different algorithms +1st test image +2nd test image +3rd test image +4th test image +5th test image +6th test image +AAGD +0 +0 +7.4627e-5 +1.3412e-5 +0.0019 +0 +LoG +0 +0 +7.4627e-5 +5.3648e-5 +0 +0 +TopHat +0 +0 +0.0044 +1.7436e-4 +0 +0 +PCM +0 +0 +0 +1.3412e-5 +0 +0 +12 + +Figure 15: Pre-thresholding results of the algorithms under the test on real infrared images (Target region is +marked by yellow circle). From the left: +the first column: original images, the second column: filtering +results of AAGD algorithm, the third column: filtering results of Tophat transform, the fourth column: +filtering results of LoG algorithm, the fifth column: filtering results of PCM algorithm. +shows poor detection performance in the 5th test image (Fig. 15). As shown in Fig. 16e, the new metrics is +completely consistent with the visual and qualitative results (Fig. 15). +6 +Conclusion +The development of new algorithms for infrared small target detection is attracted more attention during +the last decade. However, many of these recently developed algorithms do not meet the requirements of the +practical applications. Also, there are some disadvantage regarding the common evaluation metrics. In order +to completely understand the requirements of the effective evaluation metrics, the practical procedure of small +target detection should be revealed. The thresholding operation has a great role in this procedure. Without +13 + +10-1 +100 +Normalized Adjustment Parameter +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +False Alarm Rate (P +fa) +AAGD +LoG +TopHat +PCM +(a) +10-1 +100 +Normalized Adjustment Parameter +10-5 +10-4 +10-3 +10-2 +10-1 +100 +False Alarm Rate (P +fa) +AAGD +LoG +TopHat +PCM +(b) +10-1 +100 +Normalized Adjustment Parameter +10-5 +10-4 +10-3 +10-2 +10-1 +100 +False Alarm Rate (P +fa) +AAGD +LoG +TopHat +PCM +(c) +10-1 +100 +Normalized Adjustment Parameter +10-5 +10-4 +10-3 +10-2 +10-1 +100 +False Alarm Rate (P +fa) +AAGD +LoG +TopHat +PCM +(d) +10-1 +100 +Normalized Adjustment Parameter +10-5 +10-4 +10-3 +10-2 +10-1 +100 +False Alarm Rate (P +fa) +AAGD +LoG +TopHat +PCM +(e) +10-1 +100 +Normalized Adjustment Parameter +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +False Alarm Rate (P +fa) +AAGD +LoG +TopHat +PCM +(f) +Figure 16: The normalized (Pfa – k) curve in logarithmic scale. The detection performance characteristics +curve for: a) the 1st test image, b) the 2nd test image, c) the 3rd test image, d) the 4th test image, e) the 5th +test image, f) the 6th test image. +a proper thresholding strategy, the previous efforts in target enhancement algorithm development would +become obsolete. It has been demonstrated that the local statistics-based thresholding is not an appropriate +option for the segmentation of saliency map, and the the global statistics-based threshold operation is better +choice. +By considering the global thresholding as final step for the detection algorithm, the signal to clutter ratio +(SCR) metric is modified for better detection ability reflection. Also, three post-thresholding metrics are +proposed to complete performance evaluation of different algorithms. +References +[1] M. Chudecka, A. Dmytrzak, K. Le´znicka, and A. 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Bondaryk, “Morphology-based algorithm for point target +detection in infrared backgrounds,” in Signal and Data Processing of Small Targets 1993, vol. 1954. +International Society for Optics and Photonics, 1993, pp. 2–11. +[40] R. Hu, X. Zhou, G. Zhang, and G. Zhang, “Infrared dim target detection based on character filter,” +in MIPPR 2011: Automatic Target Recognition and Image Analysis, T. Zhang and N. Sang, Eds., vol. +8003, International Society for Optics and Photonics. +SPIE, 2011, pp. 319 – 325. +[41] Y. Wei, X. You, and H. Li, “Multiscale patch-based contrast measure for small infrared target detection,” +Pattern Recognition, vol. 58, pp. 216–226, 2016. +[42] H. Deng, X. Sun, M. Liu, C. Ye, and X. Zhou, “Infrared small-target detection using multiscale gray +difference weighted image entropy,” IEEE Transactions on Aerospace and Electronic Systems, vol. 52, +no. 1, pp. 60–72, 2016. +17 + diff --git a/E9E2T4oBgHgl3EQfSweS/content/tmp_files/load_file.txt b/E9E2T4oBgHgl3EQfSweS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9cc77cbc5f6abbe5e2723fd08ad2d4ce29600b6d --- /dev/null +++ b/E9E2T4oBgHgl3EQfSweS/content/tmp_files/load_file.txt @@ -0,0 +1,934 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf,len=933 +page_content='Assessing the applicability of common performance metrics for real-world infrared small-target detection Saed Moradi1, Alireza Memarmoghadam1, Payman Moallem∗1, and Mohamad Farzan Sabahi1 1Department of Electrical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, Iran Abstract Infrared small target detection (IRSTD) is a challenging task in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' During the last two decades, researchers’ efforts are devoted to improving detection ability of IRSTDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Despite the huge improvement in designing new algorithms, lack of extensive investigation of the evaluation metrics are evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Therefore, in this paper, a systematic approach is utilized to: First, investigate the evaluation ability of current metrics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Second, propose new evaluation metrics to address shortcoming of common metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' To this end, after carefully reviewing the problem, the required conditions to have a successful de- tection are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Then, the shortcomings of current evaluation metrics which include pre-thresholding as well as post-thresholding metrics are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Based on the requirements of real-world systems, new metrics are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Finally, the proposed metrics are used to compare and evaluate four well- known small infrared target detection algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The results show that new metrics are consistent with qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Keywords: Infrared small target detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' thresholding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' pre-thresholding metrics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' post-thresholding metrics 1 Introduction Nowadays, infrared (IR) imaging has a wide range of application from medical [1, 2] and industrial diagnosis [3] to defense [4] and remote sensing [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Generally, processing IR images is a challenging task [6] due to the specifications of IR imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Among all the aforementioned applications, IR small target detection (IRSTD) is a highly challenging research field because: Since the IR small targets are far from the imaging device, the target has low local contrast and appears as a dim spot in the image plane [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The small target in IR images typically occupies handful of pixels [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Thus, the region of interest (ROI) does not represent distinguished features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The edges of the small target are blurred due to atmospheric thermal fields [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Therefore, there are not a clear boundary between background area and target pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The block diagram of a typical IRSTD pipeline is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' As shown in the figure, the input IR image is first process by the IRSTD algorithm to create a saliency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Note that, while the IRSTD algorithm may refer to the end-to-end IR image processing pipeline, here, the process of construction of saliency map from input IR image is called IRSTD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The goal is to suppress the background area and enhance target pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' An ideal saliency map should eliminate the background intensities and only preserve the target area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' After saliency map reconstruction, a thresholding strategy is chosen to be applied on the saliency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Then, true (logical one) pixels in the resulting binary image is considered as target-like objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Considering the pipeline in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 1, specific attributes are defined for images in pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' In IRSTD terminology, the input image can be represented by two attributes: ∗corresponding author: p moallem@eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='ir 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='03796v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='CV] 10 Jan 2023 Input infrared image Saliency map Target-like candidates IRSTD algorithm Thresholding Input attributes Pre-thresholding attributes Post-thresholding attributes SCRin , σb,in SCRout , σb,out Pfa , Pd Figure 1: The block diagram of a typical IRSTD pipline σb,in which denotes the standard deviation of background pixels in input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' This parameter directly related to the background complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Smaller σb,in represents smooth backgrounds, while larger σb,in belongs to a complicated background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' SCRin stands for signal to clutter ratio in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' SCR is defined as µt−µb σb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Where, µt, µb, and σb are the mean value of target pixels, mean value of the local background pixels, and standard deviation of the local background, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Same argument is valid for saliency map (The processed image by IRSTD algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Thus, just like the input attributes, σb,out and SCRout represents the background complexity and the signal to clutter ration in the saliency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' It is clear that for a typical IRSTD pipeline: σb,out < σb,in and SCRout > SCRin (1) According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 1, two performance metrics are defined for evaluation of IRSTD algorithms: Background suppression factor (BSF) and signal to clutter ration gain (SCRG) which are defined as follows [10]: BSF = σb,in σb,out , SCRG = SCRout SCRin (2) Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 2, larger values for both SCRG and BSF are desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Note that, in case of evaluation of different IRSTD algorithms, since the input images are the same for all baseline algorithms, SCRout and 1 σb,out can be used as performance metrics, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The IRSTD algorithms are well-studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Mainly, these algorithms can be categorized based on filtering method, contrast measure calculation, and data structure decomposition [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The filtering based methods are divided into two sub-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The first one is the spatial domain filtering, in which, the input infrared image is processed using local kernels to enhance the target area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Max- mean [12], max-median [12], bilateral filtering [13], morphological operators [14], two dimensional least mean square [15] are some instances of this sub-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The second one refers to processing in the transformation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' In these techniques, the input image is transformed to a desired transformation space like as frequency [16] and wavelet [17] domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Then, after processing the transformed information, the inverse transform is applied to recover true targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Methods based on human visual systems (HVS) which lead to local contrast-based mechanism has received researchers’ attention during last few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' These methods outperform filter-based methods in terms of SCRG and BSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, they usually have higher computational complexity compared to filter-based ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Generally, local contrast can be constructed in either difference or ratio forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Difference local contrast like as Laplacian of Gaussian (LoG) [18], difference of Gaussian (DoG) [19], improved difference of Gabor [20], center-surround difference measure [21], and local difference adaptive measure [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Unlike the difference form local measures, ratio-form local measures utilize enhancement factor which is the ration between the center cell and surrounding ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Local contrast measure (LCM) [23], improved local contrast measure (ILCM) [24], 2 relative local contrast measure (RLCM) [25], Tri-Layer local constrast method (TLLCM) [26], novel local contrast descriptor (NLCD) [27], and weighted strengthened local contrast measure (WSLCM) [28] are the most effective IRSTDs in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' There is also a combined local measure which benefits from both difference and ratio from of local contrast measure [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Data structure decomposition-based methods are also a newly introduced class of IRSTDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Sparse and low-rank matrices decomposition is the principal of these class of IRSTDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Infrared patch image (IPI) model [30], weighted infrared patch image (WIPI) model [31], non-negative infrared patch image model based on partial sum minimization of singular values (NIPPS) [32], nonconvex rank approximation minimization (NRAM) [33], and nonconvex optimization with an Lp norm constraint (NOLC) [34] are the recent efforts of IR image decomposition-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The goal of all aforementioned methods is to obtain larger BSF and SCRG values .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, having larger BSF and SCRG does not guarantee a successful detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' A high performance IRSTD algorithm should be followed by a proper thresholding strategy to detect real targets and eliminate false responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' This is why there are two more performance metrics after applying the thresholding operation to the saliency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' These two metrics which demonstrate the ability of detection true targets and eliminating false responses are called probability of detection Pd and probability of false alarms Pfa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' In contrast to BSF and SCRG which are measurable before applying the threshold (This is why we call them pre-thresholding attributes), these two metrics are measured on binary images and therefore we call them post-thresholding attributes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' As mentioned in the previous paragraph, for a successful detection, both high performance IRSTD al- gorithm as well as the proper thresholding strategy are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Regardless of effectiveness of the IRSTD algorithm, improper thresholding will leads to missing true targets and having false responses which could be disaster for a practical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Hence, in this paper, after investigating various thresholding strategies, the best methods for applying threshold to the saliency map is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Then, current pre-thresholding as well as the post-thresholding metrics are investigated, and some new metrics which are aligned with practical considerations are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The rest of this paper is organized as follows: in the next section, the role of thresholding in practical systems is deeply investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Then, in section 3, current pre-thresholding metrics are reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' After demonstrating their shortages, modified metrics are proposed for IRSTD performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' In section 4, same process is performed for post-thresholding metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' In section 5, The newly proposed metrics are used for performance comparison of common IRSTD algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Finally, the paper is concluded in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 2 The onus of thresholding on the overall performance After performing target enhancement and clutter suppression procedure (saliency map construction), the filtered IR image should be converted to binary one using thresholding operation that can be applied in different forms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='e manual, automatic, local, global).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Since the target detection problem only consists of two different classes namely as target and background clutter, single-level thresholding is a satisfactory option for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The simplest method to achieve the classification goal, is to apply a global threshold T: g(x, y) = � 1 f(x, y) > T 0 f(x, y) ≤ T (3) where, f(x, y) and g(x, y) stand for the saliency map and binary image,respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The most challenging part of global thresholding operation is how to set an effective threshold value T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Since target detection systems continuously scan the environment, human operator cannot be helpful to choose the optimum threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The most simplest way to do this is to choose a unique threshold value based on experiments for all incoming image frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, when the dynamic range of the filtered image is not equal to the dynamic range of the input images, the false-alarm rate or the miss-rate will increase drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 2 shows the change in the dynamic range of filtered images (saliency maps) using Tophat and AAGD IRSTDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' As shown in the figure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' the output dynamic range directly depends on the applied IRSTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Therefore, the thresholding procedure should be performed in an automated manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' There are various automatic image thresholding algorithms in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The Otsu’s method is one of the widely used one [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' In this method the global threshold value is chosen in a way, to maximize inter-class variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' When both foreground (Target) and 3 40 60 80 100 120 140 160 180 200 (a) 0 10 20 30 40 50 60 70 80 90 100 (b) 0 500 1000 1500 2000 2500 (c) Figure 2: Variable dynamic range in saliency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' a) Original infrared image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' b) filtering result using TopHat algorithm [14], c) filtering result using AAGD [37] algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The dynamic range of the saliency map might be different than input infrared image depending on the applied IRSTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' background classes include considerable number of pixels, and the image histogram is bimodal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' there is a deep valley between two peaks in the image histogram), the Otsu’s method works very well in object segmentation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, when the target area is too small compared to the background area, which is always occurred in incoming infrared target detection problems, the segmentation result of Otsu’s method is inaccurate Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Another widely used automatic thresholding is presented in [36], where the followings are performed to obtain the desired threshold value: i) The gray image is segmented into two classes using threshold value equal to global mean of the image (T = µG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' ii) The average values of the background and target are calculated (µB, µT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' iii) The new threshold level is calculated � Tnew = 1 2 (µT + µB) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' iv) While (Tnew − Told > ϵ), steps (ii) and (iii) are recursively repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' When the background noise is not strong, this automatic thresholding operation shows good performance for final target detection (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, in strong noisy scenarios, the performance of this algorithm is degraded significantly (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 4d), which in turn, increases the false responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Moreover, when infrared scenario does not contain any small target, these histogram-based automatic thresholding methods always return incorrect responses in non-target areas (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Statistics-based image thresholding is the most effective thresholding strategy for small target detection which can be applied in both local and global manners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Statistics-based global and local thresholding are expressed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 4 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' T = µG + kG × σG (4) T(x, y) = µ(x, y) + kL × σ(x, y) (5) where, µG, σG, µ(x, y), σ(x, y), kG, and kL indicate global mean of the image, global standard deviation of the image, local mean around (x, y) position, local standard deviation around (x, y) position, control parameter of global thresholding and local thresholding, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Global thresholding is a simple operation with low computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, in multi-target scenarios, some targets may be missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Local thresholding can detect all targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Since local mean and standard deviation should be calculated for each pixel in the gray image, the local statistics-based thresholding has higher computational complexity compared to the global one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Generally speaking, using statistics-based thresholding has the following advantages: It can work with any gray-level dynamic ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The control parameter (k) can be determined by experiments to achieve reasonable false-alarm rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The last but not the least, it is very effective for scenarios with no targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 4 (a) (b) (c) (d) (e) Figure 3: The automatic thresholding results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' a) Original infrared image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' b) Top-hat filtering result [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' c) Otsu’s thresholding result (T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' d) automatic thresholding using average values of background and target classes (T = 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' e) Manual thresholding (T = 29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' (a) (b) (c) (d) (e) Figure 4: The automatic thresholding results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' a) Original noisy infrared image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' b) Top-hat filtering result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' c) Otsu’s thresholding result (T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' d) automatic thresholding using average values of background and target classes (T = 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' e) Manual thresholding (T = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 3 Pre-thresholding evaluation A detection process is successful as long as a single pixel of target area is correctly recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' In this case, the exact boundary extraction of the target area is not important at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Therefore, a proper evaluation metric should support this argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Signal to clutter ratio (SCR) is one of pre-thresholding metrics which shows the target enhancement 5 (a) (b) (c) Figure 5: Drawback of automatic thresholding in scenarios with no targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' a) original infrared image which does not contain small target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' b) the result of Top-Hat filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' c) automatic thresholding results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' ability of an IRSTD, which is defined as: SCR = µT − µb σb , (6) where, µT , µb, and σb denote average intensity of the target area, average intensity and standard deviation of its local surrounding background, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' While this evaluation measure is generally accepted in the lit- erature, it can not correctly reflect the target enhancement capability of an IRSTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' To better understanding, a simple scenario is provided here (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Two different saliency maps are demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6a shows the result of applying AAGD algorithm [37] with 9 × 9 internal window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' As depicted in the figure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' the target area is relatively enhanced while there are some remaining background clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Compared to the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6a, the second IRSTD which again is an AAGD Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6b algorithm with 3 × 3 internal window followed by a morphological erosion with a 3 × 3 square-shape structural element, shows better target enhancement and background suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6d, the signal amplitude for the IRSTD #2 is almost twice as the one in the IRSTD #1, which means in higher threshold values the target will be detected corectly in the second one, while in the first one the target will be missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' One dimensional (1D) cross-section of target area in both saliency maps are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6e and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' To simplify the scenario, let’s approximate 1D cross-section of the target area with closest square signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' As shown in the figure: AT1 = AT2 2 , WT1 = 3 × WT2 (7) where, AT1, AT2 denote the target amplitude in the output of IRSTD #1 and #2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Also, WT1, WT2 show the target width (extension) in the output of IRSTD #1 and #2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Based on SCR formulation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6), and simply considering zero-mean background signal, the following relationship can be easily derived: SCR1 = 3 2 × SCR2 (8) which implies that the target detection ability of the IRSTD #1 is 50% more than that of IRSTD #1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, by applying a global threshold level at Tapp, the #1 algorithms does not detect the true target (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' It can be clearly seen that the #2 algorithm can detect the true target at the same threshold level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' In order to address this issue when global thresholding is final choice in practical system, the SCR formulation should be modified as: SCRglobal = maxT −µG σG , (9) where, maxT denotes the maximum gray value of the target area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 4, the maximum acceptable control parameter is equal to newly defined SCR metric: kGmax = SCRglobal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' (10) There are two important points regarding the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 10: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='6 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='450 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='(f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Threshold Value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Tapp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Signal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Amplitude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='AT1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='WT1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Signal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Output of #1 detection algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Threshold Value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Tapp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Signal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Amplitude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='AT2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='WT2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Signal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Output of #2 detection algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='(g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Figure 6: A simple scenario to demonstrate the drawback of common SCR metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' a, b) Saliency map of the IRSTD #1 and #2, c, d) target area in the saliency map of the IRSTD #1 and #2, e, f) 1D plot of target cross-section in c and d, g) simplified 1D representation of target area in both IRSTD #1 and #2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Common SCR metric is not able to correctly reflect the target detection ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The pre-thresholding evaluation should be performed in a global manner on the saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The thresholding operation should be consistent with pre-thresholding evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' For instance, in our case, the global statistics-based thresholding is the right choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' So far, it is demonstrated that the global thresholding is the right one to be applied on the saliency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' In the next subsection, we demonstrate the drawback of the local statistics-based thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='1 Drawback of common local thresholding Now, let consider the case that local thresholding is supposed to be applied on the saliency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6, the local mean around target region can be calculated as follows: µ(x) = AW n (11) where, A, W, x, and n stand for the target amplitude, width (spatial extension), the current index and number of samples in local neighborhood (n > W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 7 2 3 4 5 6 7 8 9 1 2 3 4 kL max n=17 n=25 n=33 Figure 7: kLmax versus target spatial extension W (target width in 1D case) The local standard deviation can be calculated as: σ(x) = � � � � 1 n n � i=1 (y(i) − µ(x))2 (12) Where y(i) and µ(x) denote the saliency map samples and local mean, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Since the detection algorithm is supposed to suppress background clutter, for the sake of simplicity, we can assume that the saliency map samples out of the target region are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Then: σ(x) = A n �� nW − W 2 � (13) The local thresholding (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 5), can be rewritten as: T(x) = µ(x) + kL × σ(x) = AW n + kLA n �� nW − W 2 � (14) The detection process is established correctly for the threshold values lower than target amplitude (T(x) < A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Therefore, for a successful target detection the following condition should be met: µ(x) + kL × σ(x) < A ⇒ AW n + kLA n �� nW − W 2 � < A (15) The upper bound for control parameter (kLmax) to detect the target accurately can be find as follows: AW n + kLmaxA n �� nW − W 2 � = A ⇒ kLmax = � n − W W (16) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 7 shows the upper bound of control parameter versus target width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' As shown in the figure, the maximum control parameter to detect target correctly using local thresholding decreases as the target width increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' kLmax takes its maximum value when the target width is equal to one pixel (kLmax = √n − 1 for W = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' This result is quite consistent with the fact that the most effective target detection algorithm should suppress all background region and only returns a single pixel (Target centroid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 8 shows the local threshold value which is normalized to the target amplitude ( T A) versus different control parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' It is clear that, only when the ( T A) fraction is less than one the target can be detected correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Another finding which can be derived from the figure is that the maximum control parameter decreases as the target width increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Therefore, unlike the global case, the effective control parameter to extract real targets and eliminate background clutter depends on the target area in the saliency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Also, the reasonable rang for control parameter is narrowed when the local neighborhood is decreased (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Moreover, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 5 to extract true target from saliency map leads to many false alarms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 9 shows the 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5 3 Local Adjustment Parameter (KL) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5 W=3 W=5 W=7 W=9 The target is missed The target is correctly detected (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5 3 Local Adjustment Parameter (KL) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5 2 W=3 W=5 W=7 W=9 The target is missed The target is correctly detected (b) Figure 8: Local threshold value normalized to target amplitude ( T A) versus different control parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' a) n = 33, b) n = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 9: Shortcoming of local thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' a) Original input image, the target area is marked by red ellipse, b) the result of target enhancement using multi-scale Laplacian of Gaussian (LoG) method, c) the local thresholding applied on (b) with k = 4, c) the local thresholding applied on (b) with k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' local thresholding on the saliency map of multi-scale Laplacian of Gaussian (LoG) method [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' As shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 9b, the target area is the most salient region in saliency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, after thresholding using local method (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 9c), there are too many false responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The only way to limit false responses to an acceptable range is to increase the control parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, the true target is not extracted when the local threshold is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Note that there are still too many false responses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 9d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Based on the local thresholding results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 9, this method (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 5) is not a proper strategy to discriminate target area from background clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 4 Post-thresholding evaluation After applying a predefined threshold to the saliency map, a binary image is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' In this case, the prevalent metrics to evaluate the performance of the detection algorithms are probability of false-alarm Pfa and detection Pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' These two metrics are defined as [38]: Pfa = Nf Ntot , Pd = Nd Nr (17) 9 10 −10 10 −5 10 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='8 1 Pfa Pd Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 1 Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 2 Figure 10: ROC curve for two typical detectors (a) (b) (c) Figure 11: a) original image, b) the LCM filtering result, c) the Top-Hat filtering result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='9 1 Pfa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='95 1 PD LCM TopHat (a) 0 20 40 60 80 100 120 140 160 180 200 T 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='9 1 Pfa LCM TopHat (b) Figure 12: a) The ROC curve, b) false-alarm rate versus different threshold levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' where Nf, Ntot, Nd, and Nr denote the number of wrongly detected pixels, the total number of pixels, the number of pixels which are detected correctly, and the target pixels in the ground-truth, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The receiver operational characteristics (ROC) curve is constructed by considering each (Pfa, Pd) pair at different threshold level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 10 shows the ROC curve for two typical detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' As shown in the figure, for a constant false-alarm rate, the detector #1 has higher detection rate, and outperforms the algorithm #2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The ROC curve is a satisfactory tool to evaluate the performance of different detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, if the detection rate and false- alarm rate are not defined accurately, the final ROC curve is not a reliable measure anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' In order to demonstrate the deficiency of the definitions of the Pd and Pfa (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 17), let consider the target detection ability of two well-known small infrared target detection algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Local contrast method (LCM) [23] and Top-hat algorithm [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 11 shows the detection results of these two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' As shown in the figure, the Top-hat filtering method clearly outperforms the LCM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' however, the ROC curve gives contradictory result against visual perception (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 12a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Also, by constructing the curve of the false-alarm rate versus different threshold levels (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 12b), the low performance of the LCM algorithm is clearly seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Therefore, the former definition of the Pd (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 17) is not appropriate for this crucial metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Another alternative definition for Pd is suggested in the literature ([25]): Pd = ND NR (18) where ND and NR are number of detected true targets, and total number of true targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' While this new definition addresses the deficiency of the former one (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 17), there are still some drawbacks regrading this formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The real infrared scenarios usually contain limited number of targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' To overcome this drawback, 10 (a) (b) (c) (d) Figure 13: a) Synthetic targets in homogeneous local background, b) low contrast targets in background clutter edges, c) the character filter response to a and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 14: a, c) real infrared scenario, b , d) the response of character filter [40] to a and c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' synthetic targets are usually created using Gaussian spatial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, spatial distribution-based target detection algorithms directly benefit from synthetic data, so the final evaluation is not fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' An example is provided here to better demonstration of this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The character filter [40] utilizes Gaussian spatial distribution as a measure to distinguish between real target and background clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 13, when the small target has exactly Gaussian distribution, the character filter effectively can enhance the small targets and eliminate background clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, in real infrared scenarios, which the spatial distribution of small targets does not follow the Gaussian distribution [38], the detection results of character filter is chaotic (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' According to aforementioned issues regarding the post-thresholding performance evaluation metrics, herein, awe present a new approach capable of addressing all the shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Since in a successful de- tection operation, at least one pixel is detected after thresholding operation, the following procedure is introduced to obtain new post-thresholding performance measure: i) The upper bound for control parameter (kmax) is calculated (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' ii) The [0 − kmax] interval is chosen as valid interval for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' iii) For each different control parameters, the false-alarm rate is calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 11 Table 1: The baseline algorithms Detection Algorithm Details Top-Hat [39] 7 × 7 structural element LoG [18] With [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='72, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='86, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='03, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='24, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='49, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='79, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='14, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='57, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='09, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='71] scale parameters PCM [41] With [3 × 3, 5 × 5, 7 × 7, and 9 × 9] cell-sizes AAGD [42] With [3 × 3, 5 × 5, 7 × 7, and 9 × 9] cell-sizes Table 2: The value of maximum control parameter kmax for different algorithms the 1st test image the 2nd test image the 3rd test image the 4th test image the 5th test image the 6th test image AAGD 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='8553 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='9809 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='8804 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='9033 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='0117 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='4851 LoG 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5312 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='8976 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='4743 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='2515 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='4517 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='4031 TopHat 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='8858 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='8994 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5513 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='8742 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='4408 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='9160 PCM 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='7368 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='1190 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='5538 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='4819 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='4008 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='0127 iv) The false-alarm rate versus control parameter (Pfa – k) curve is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' In the next step, the [0 – k] interval is linearly mapped to [0 – 1] range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' This normalization allows us to fairly compare and evaluate different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' After constructing (Pfa – k) curve, the following measures can be extracted: The maximum control parameter (kmax) is the first inferred performance evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The larger kmax, the higher detection ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The false-alarm rate at kmax, which is called Pfa,min here, is the second evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' It is obvious that the false-alarm rate of the system can not be less than Pfa,min while the true target is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' After normalizing [0 – k] interval to [0 – 1] range, the false alarm rate of the detection algorithms can be plotted in single figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Then, the algorithm with satisfying detection performance can be chosen for the practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 5 Detection ability evaluation using new metrics In order to evaluate the detection ability using the proposed metrics, four well-known small infrared target detection algorithms are chosen to conduct the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 1 reports the baseline algorithms and their implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The pre-thresholding enhancement results of each algorithm are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Visually speaking, the AAGD algorithm has better performance in background suppression (the background region is mapped to zero value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, the most part of gray area in PCM output have zero values (Since there are also negative values in the saliency map, the zero values are depicted by gray color instead of black one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' LoG and TopHat filters are sensitive to noise and sharp edges, therefore, there are too many false responses in their saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The results of evaluation using new metrics are reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 2 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' As reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 2, AAGD and PCM algorithms have better enhancement for target area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, by taking the false-alarms into account, the PCM algorithm shows better clutter rejection ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 16 shows the normalized (Pfa – k) curve to investigate the detection performance charac- teristics of different baseline algorithms, and fairly compare them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' As shown in the figure, the PCM and AAGD algorithm has overall superiority compared to LoG and TopHat algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The AAGD algorithm Table 3: The value of minimum probability of false alarm Pfa,min for different algorithms 1st test image 2nd test image 3rd test image 4th test image 5th test image 6th test image AAGD 0 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='4627e-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='3412e-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='0019 0 LoG 0 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='4627e-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='3648e-5 0 0 TopHat 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='0044 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='7436e-4 0 0 PCM 0 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='3412e-5 0 0 12 Figure 15: Pre-thresholding results of the algorithms under the test on real infrared images (Target region is marked by yellow circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' From the left: the first column: original images, the second column: filtering results of AAGD algorithm, the third column: filtering results of Tophat transform, the fourth column: filtering results of LoG algorithm, the fifth column: filtering results of PCM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' shows poor detection performance in the 5th test image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 16e, the new metrics is completely consistent with the visual and qualitative results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' 6 Conclusion The development of new algorithms for infrared small target detection is attracted more attention during the last decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' However, many of these recently developed algorithms do not meet the requirements of the practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Also, there are some disadvantage regarding the common evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' In order to completely understand the requirements of the effective evaluation metrics, the practical procedure of small target detection should be revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The thresholding operation has a great role in this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' Without ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Normalized Adjustment Parameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='False Alarm Rate (P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='fa) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='AAGD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='LoG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='TopHat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='PCM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Normalized Adjustment Parameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='False Alarm Rate (P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='fa) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='AAGD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='LoG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='TopHat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='PCM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Normalized Adjustment Parameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='False Alarm Rate (P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='fa) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='AAGD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='LoG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='TopHat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='PCM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Normalized Adjustment Parameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='False Alarm Rate (P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='fa) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='AAGD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='LoG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='TopHat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='PCM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Normalized Adjustment Parameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='False Alarm Rate (P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='fa) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='AAGD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='LoG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='TopHat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='PCM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Normalized Adjustment Parameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='False Alarm Rate (P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='fa) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='AAGD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='LoG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='TopHat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='PCM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='(f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content='Figure 16: The normalized (Pfa – k) curve in logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' The detection performance characteristics curve for: a) the 1st test image, b) the 2nd test image, c) the 3rd test image, d) the 4th test image, e) the 5th test image, f) the 6th test image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' a proper thresholding strategy, the previous efforts in target enhancement algorithm development would become obsolete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E2T4oBgHgl3EQfSweS/content/2301.03796v1.pdf'} +page_content=' It has been demonstrated that the local statistics-based thresholding is not an appropriate option for the 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WATER WAVES OF FINITE DEPTH +JON WILKENING AND XINYU ZHAO +Abstract. We present a numerical study of spatially quasi-periodic water waves of finite depth in +both the initial value problem and traveling wave settings. We adopt a quasi-periodic conformal +mapping formulation of the Euler equations, where one-dimensional quasi-periodic functions +are represented by periodic functions on a higher-dimensional torus. We compute the time +evolution of free surface waves in the presence of a background flow and a quasi-periodic +bottom boundary and observe the formation of quasi-periodic patterns on the free surface. +Two types of quasi-periodic traveling waves are computed: small-amplitude waves bifurcating +from the zero-amplitude solution and larger-amplitude waves bifurcating from finite-amplitude +periodic traveling waves. We derive weakly nonlinear approximations of the first type and +investigate the associated small-divisor problem. We find that waves of the second type exhibit +striking nonlinear behavior, e.g., the peaks and troughs are shifted non-periodically from the +corresponding periodic waves due to the activation of quasi-periodic modes. +1. Introduction +Free surface waves on incompressible fluids arise in many contexts in fluid dynamics. Ex- +amples include ocean wave forecasting [38, 61], modeling the motion of flows over obstacles +and varying bottom boundaries [5, 29, 67], and studying wind-wave interactions in extreme +wave events, such as freak waves [40]. These models are described by the Euler equations, +which are usually studied under periodic boundary conditions or the assumption that solu- +tions decay to zero at infinity [4,33,39]. However, these assumptions are insufficient in many +problems of interest. For instance, a periodic wave could interact with a bottom boundary with +a different spatial period, or subharmonic perturbations of a periodic traveling wave can grow +in amplitude, leading to quasi-periodic waves. To tackle these issues, we recently proposed +methods [73, 74] to study the Euler equations under quasi-periodic boundary conditions; +specifically, we studied spatially quasi-periodic waves of infinite depth in two dimensions +and developed numerical algorithms to compute such waves. In this paper, we extend this +previous work to the finite-depth case and discuss both the initial value and traveling wave +problems in the quasi-periodic setting. +Finite-depth water waves exhibit interesting nonlinear dynamics. It has been shown nu- +merically that Fermi-Pasta-Ulam recurrence can occur in free surface waves of finite depth +when the wave amplitude is less than about 1{10 of the fluid depth [56,58]. A varying bottom +boundary can lead to substantial amplifications of water waves. There have been both ex- +perimental and numerical studies demonstrating increased freak wave activities when waves +propagate over a sloping bottom, from a deeper to a shallower domain [21,64]. In the problem +of long waves approaching vertical walls, an abrupt transition in the bottom boundary can +cause large runups on the wall or wave breaking, which generally occurs when the wave crest +Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, USA +Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1 +E-mail address: wilkening@berkeley.edu, zhaox171@mcmaster.ca. +1 +arXiv:2301.01289v1 [physics.flu-dyn] 3 Jan 2023 + +2 +J. WILKENING AND X. ZHAO +overturns [34, 66]. The interaction between a rotational wave current and a varying bottom +boundary gives rise to a time-dependent Kelvin cat-eye structure [29]. +The quasi-periodic dynamics of water waves have recently drawn considerable attention. +Damanik and Goldstein [18] proved the global existence and uniqueness of small-amplitude +spatially quasi-periodic solutions of the KdV equation. Oh [51] and Dodson et al. [20] showed +the local existence of spatially quasi-periodic solutions of nonlinear Schrödinger equations. +Berti and Montalto [10] and Baldi et al. [7] used Nash-Moser theory to prove the existence +of small-amplitude temporally quasi-periodic gravity-capillary and gravity standing waves. +Berti et al. [8, 9] and Feola and Giuliani [27] have proved the existence of temporally quasi- +periodic gravity-capillary and gravity waves. On the numerical side, Wilkening computed +new families of relative-periodic [70] and traveling-standing [71] water wave solutions. +We were originally motivated by the structure of quasi-crystals in material science. Bli- +nov [11] used quasi-periodic solutions of the Schrödinger equation to describe the electronic +structure of non-interacting electrons of quasi-crystals. To study how electrons move through +quasi-crystals, Torres et al. [62] created quasi-periodic standing waves by vibrating a fluid- +filled pan with a quasi-periodic bottom boundary and sent a transverse wave pulse across +the fluid. They observed that the traveling wave pulse demonstrated a non-periodic pattern: +the spacing between the wave peaks was not constant. Their observation inspired us to ask +the following question: how do we compute the exact dynamics of free surface waves in the +presence of a quasi-periodic bottom boundary? To address this question, one needs to study +the free surface wave problem in a quasi-periodic framework. +Another reason for our interest in quasi-periodic water waves originates from the dispersion +relation of gravity-capillary waves of finite depth: +(1.1) +푐2 “ p푔푘´1 ` 휏푘q tanhp푘ℎq. +Here 푐 is the phase speed, 푘 is the wave number, 푔 is the acceleration due to gravity, 휏 is +the coefficient of surface tension and ℎ is the depth of the fluid. It is known [63] that when +휏{p푔ℎ2q ă 1{3, there exists 푐crit between 0 and +a +푔ℎ such that for any fixed phase speed +푐 ą 푐crit, there are two distinct positive wave numbers satisfying the dispersion relation (1.1), +which we denote by 푘1 and 푘2. Any superposition of waves with these two wave numbers +is a solution of the linearized traveling wave problem. If 푘1 and 푘2 are rationally related, the +linear solution is spatially periodic and related to the well-studied Wilton ripples [1,2,63,76]. +On the other hand, if 푘1 and 푘2 are irrationally related, the linear solution will be spatially +quasi-periodic, which gives a natural place to search for nonlinear quasi-periodic traveling +solutions. Bridges and Dias [14] first studied these spatially quasi-periodic traveling waves +using a spatial Hamiltonian structure and constructed weakly nonlinear approximations of +these waves. Later we [73] used a conformal mapping formulation of the water wave equations +and computed highly accurate numerical solutions of the fully nonlinear problem in the case +of deep water. In the present work, we aim to further extend these techniques to the case of +finite-depth water. +Following [73, 74], we adopt a conformal mapping formulation of the free surface Euler +equations [16,22–25,36,42,78]. In the finite depth case, the fluid domain with a curved surface +and an uneven bottom boundary is mapped conformally onto a horizontal strip instead of +the lower half-plane. Since the conformal mapping depends on time, even though the bottom +boundary is fixed in physical space, the representation of the bottom boundary in conformal +space varies with time. Ruban [55, 57] fixed the width of the strip and used a composition +of two conformal mappings to map the strip to the fluid domain – the first leaves the real + +QUASI-PERIODIC WATER WAVES OF FINITE DEPTH +3 +axis invariant and the second maps the real line to the bottom boundary. Viotti et al. [67] +and Flamarion et al. [30] let the width of the strip vary with time to keep the wave length +the same in physical space and conformal space. They used a fixed-point iterative method +to compute the bottom profile at different times. +In order that water waves possess the +same quasi-periods in both physical and conformal spaces, we also let the strip width be a +time-dependent variable. However, in contrast to [30, 67], we compute the time evolution +of the bottom profile directly, employing analytical properties of the conformal mapping, +similar to [55, 57]. As in the infinite-depth case [73, 74], we introduce quasi-periodic Hilbert +transforms to relate the real and imaginary parts of the conformal mapping and to compute +the kinematic boundary condition on the free surface. These Hilbert transforms are Fourier +multiplier operators and are easier to compute in a quasi-periodic setting than a more direct +computation of the Dirichlet-Neumann operator [17] in physical space, e.g., using boundary +integral methods [5]. +In computing the dynamics of free surface waves over a varying bottom boundary, it is +usually assumed that the spatial periods of the free surface wave and the bottom boundary +are the same or one is an integer multiple of the other. +In this paper, we study a new +situation where their spatial periods are irrationally related. Specifically, in one of the examples +presented in Section 4.1, we compute the time evolution of an initially periodic free surface +wave with period 2휋 in the presence of a periodic bottom boundary with period 2 +? +2휋. We +find that the periodic wave becomes a quasi-periodic wave, with each wave peak and trough +evolving differently as it interacts with the bottom boundary. By the time-reversibility of the +Euler equations, we learn that a quasi-periodic wave can evolve to a periodic wave. We also +compute the time evolution of an initially flat free surface in the presence of a background flow +and a quasi-periodic bottom boundary. Similar to the experiment by Torres et al. [62], we also +observe that the free surface wave develops quasi-periodic patterns as a result of interactions +between the background flow and the quasi-periodic bottom boundary. The wave peaks and +troughs are asymmetric and the distance between adjacent wave peaks is not constant. +In Section 4.2, we compute two types of quasi-periodic traveling solutions: waves that +bifurcate from the zero-amplitude solution and waves that bifurcate from finite-amplitude +periodic traveling solutions. For the first type, we use linearization about the zero solution for +the initial bifurcation direction and obtain a three-parameter family of solutions prescribed by +the fluid depth and Fourier coefficients corresponding to wave numbers 푘1 and 푘2; these are +called the base Fourier coefficients. Similar to the case of deep water [73], when the amplitudes +of the base Fourier coefficients are small, the solutions are of small amplitude and are close +to the linear solution. For the second type, we linearize the governing equations around a +finite-amplitude 2휋-periodic traveling wave. For the bifurcation direction in this case, we use +a quasi-periodic function of the following form in the kernel of the linear operator: +(1.2) +훿휂p훼q “ 푒푖푘훼휂0p훼q ` 푐.푐., +where 휂0 possesses the same wavelength as the periodic traveling wave, the notation 푐.푐. +denotes the complex conjugate of the preceding term, and we set 푘 “ 1{ +? +2 in this paper. +This method has also been used to compute secondary periodic bifurcations with 푘 “ 1{2 and +푘 “ 1{3 by Chen and Saffman [15] and with 푘 “ 1{9 by Vanden-Broeck [65]. In the present +work, we obtain quasi-periodic traveling waves that bifurcate from a periodic traveling wave +whose first Fourier mode resonates with the fifth Fourier mode. The periodic traveling wave is +a solution of the Wilton ripple problem and the wave peaks look like “cat ears”. The bifurcated + +4 +J. WILKENING AND X. ZHAO +wave still preserves this characteristic; however, influenced by the Fourier modes in the quasi- +periodic direction, the distance between the successive “ears” is no longer constant. +The paper is organized as follows. +In Section 2, we define finite-depth quasi-periodic +Hilbert transforms and derive equations of motion for quasi-periodic free surface waves in +conformal space when the bottom boundary is not necessarily flat. In Section 3, we obtain +the governing equations of quasi-periodic traveling waves in the case of finite-depth water +with a flat bottom boundary and establish weakly nonlinear approximations of these waves +and the role of small divisors in computing successive approximations. In Section 4, we use a +Fourier pseudo-spectral method to compute solutions of the initial value and traveling wave +problems and present various numerical examples. Following the idea in [73,74], we lift the +one-dimensional quasi-periodic problem to a higher-dimensional periodic torus where the +computation is performed. We formulate the traveling wave problem as an overdetermined +nonlinear least-squares problem that we solve through a variant of the Levenberg-Marquardt +method [50, 72]. For the initial value problem, we consider the natural setting where the +quasi-periodic initial condition and bottom boundary are posed in physical space. We present +a method of transforming them to conformal space in Appendix A. +2. Equations of motion +2.1. Governing equations in physical space. Gravity-capillary waves of finite depth are gov- +erned by the free-surface Euler equations [39,77]. In two dimensions, they may be written +(2.1) +휂sp푥, 0q “ 휂s +0p푥q, +휑p푥, 0q “ 휑0p푥q, +푡 “ 0, +푥 P R, +(2.2) +Φ푥푥 ` Φ푦푦 “ 0, +휂bp푥q ă 푦 ă 휂sp푥, 푡q, +Φ “ 휑, +푦 “ 휂sp푥, 푡q, +∇Φ ¨ 풏 “ 0, +푦 “ 휂bp푥q, +(2.3) +휂s +푡 “ Φ푦 ´ 휂s +푥Φ푥, +푦 “ 휂sp푥, 푡q, +(2.4) +휑푡 “ Φ푦휂s +푡 ´ 1 +2Φ2 +푥 ´ 1 +2Φ2 +푦 ´ 푔휂s ` 휏 +휂s +푥푥 +` +1 ` p휂s +푥q2˘3{2 ` 퐶p푡q, +푦 “ 휂sp푥, 푡q, +where 푥 is the horizontal coordinate, 푦 is the vertical coordinate, 푡 is the time, Φp푥, 푦, 푡q is the +velocity potential of the fluid, 휂sp푥, 푡q is the free surface elevation, 휂bp푥q is the fixed bottom +profile, 푔 is the vertical acceleration due to gravity, and 휏 is the coefficient of surface tension, +which is zero for gravity waves. Equation (2.3) is the kinematic boundary condition and (2.4) +is the dynamic boundary condition. The function 퐶p푡q in (2.4) is an arbitrary integration +constant that is allowed to depend on time but not space. We are interested in the dynamics +of the water waves in the presence of a varying bottom boundary; in other words, the bottom +profile is not a constant function. When the bottom boundary is flat, it is usually assumed +that there is no background flow. Indeed, in this case, the system is Galilean invariant, which +means any background flow can be eliminated by viewing the system in a moving frame. +However, this is not true when the bottom boundary is variable; the interaction between +the background flow and the bottom boundary can lead to interesting nonlinear dynamics. +Therefore, it is meaningful to incorporate a background flow in the problem description by +including a secular growth term in the velocity potential, which is otherwise spatially periodic +or quasi-periodic. + +QUASI-PERIODIC WATER WAVES OF FINITE DEPTH +5 +2.2. Quasi-periodic Hilbert transforms. As defined in [26,46], a quasi-periodic, real-analytic +function 푓 p훼q is a function of the form +(2.5) +푓 p훼q “ ˜푓 p풌훼q, +˜푓 p휶q “ +ÿ +풋PZ푑 +ˆ푓풋푒푖x풋, 휶y, +훼 P R, 휶, 풌 P R푑, +where x¨, ¨y denotes the standard inner product on R푑 and ˜푓 is a periodic, real-analytic function +defined on the 푑-dimensional torus +(2.6) +T푑 :“ R푑L +p2휋Zq푑. +We assume that 푑 ě 2 so that 푓 can be genuinely quasi-periodic. Entries of the vector 풌 +are called the basic wave numbers (or basic frequencies) of 푓 and are required to be linearly +independent over Z. Given a quasi-periodic function 푓 , the corresponding ˜푓 and 풌 in (2.5) are +not unique. Indeed, if 푲 is any 푑-by-푑 unimodular matrix, then ˜푓 1p휶q “ ˜푓 p푲휶q also satisfies +(2.5) with 풌1 “ 푲´1풌. For simplicity, we assume 풌 is given, along with 푓 or ˜푓 , to pin down the +representation. Given 풌, one can reconstruct ˜푓 and its Fourier coefficients ˆ푓풋 from 푓 via +(2.7) +ˆ푓풋 “ lim +푎Ñ8 +1 +2푎 +ż 푎 +´푎 +푓 p훼q푒´푖x풋,풌y훼푑훼, +풋 P Z푑. +We refer to [6,12,28,35,46] for detailed discussions of the above averaging formula. We assume +that ˜푓 p휶q is real-analytic, which is equivalent to the conditions that ˆ푓´풋 “ ˆ푓풋 for 풋 P Z푑 and +there exist positive numbers 푀 and 훾 such that | ˆ푓풋| ď 푀푒´훾}풋}, i.e., the Fourier modes ˆ푓풋 decay +exponentially as }풋} Ñ 8. Next we introduce some operators that act on 푓 and ˜푓 . +Definition 2.1. The projection operators 푃 and 푃0 are defined by +(2.8) +푃 “ id ´푃0, +푃0r 푓 s “ 푃0r ˜푓 s “ ˆ푓0 “ +1 +p2휋q푑 +ż +T푑 +˜푓 p휶q 푑훼1 ¨ ¨ ¨ 푑훼푑. +Note that 푃 projects onto the space of zero-mean functions and 푃0 returns the mean value. +There are two versions of 푃 and 푃0, one acting on quasi-periodic functions defined on R and +one acting on torus functions defined on T푑. +Definition 2.2. The derivative operator B훼 that acts on 푓 or ˜푓 is defined by +(2.9) +B훼 푓 p훼q “ B훼 ˜푓 p풌훼q, +B훼 ˜푓 p휶q “ +ÿ +풋‰0 +푖x풋, 풌y ˆ푓풋푒푖x풋,휶y. +For simplicity of notation, we denote B훼 푓 (or B훼 ˜푓 ) by 푓훼 (or ˜푓훼). One can also interpret B훼 ˜푓 as +the directional derivative of ˜푓 along the characteristic direction 풌. +Definition 2.3. We introduce four quasi-periodic Hilbert transforms 퐻tanh, 퐻coth, 퐻csch, 퐻sech +that act on 푓 and ˜푓 as follows +(2.10) +퐻tanhr 푓 sp훼q “ 퐻tanhr ˜푓 sp풌훼q, +퐻tanhr ˜푓 sp휶q “ +ÿ +풋‰0 +푖 tanh +` +x풋, 풌yℎ +˘ ˆ푓풋푒푖x풋,휶y, +퐻cothr 푓 sp훼q “ 퐻cothr ˜푓 sp풌훼q, +퐻cothr ˜푓 sp휶q “ +ÿ +풋‰0 +p´푖q coth +` +x풋, 풌yℎ +˘ ˆ푓풋푒푖x풋,휶y, +퐻sechr 푓 sp훼q “ 퐻sechr ˜푓 sp풌훼q, +퐻sechr ˜푓 sp휶q “ +ÿ +풋‰0 +sech +` +x풋, 풌yℎ +˘ ˆ푓풋푒푖x풋,휶y, +퐻cschr 푓 sp훼q “ 퐻cschr ˜푓 sp풌훼q, +퐻cschr ˜푓 sp휶q “ +ÿ +풋‰0 +p푖q csch +` +x풋, 풌yℎ +˘ ˆ푓풋푒푖x풋,휶y. + +6 +J. WILKENING AND X. ZHAO +Figure 1. The time-dependent conformal mapping. +Here ℎ is a positive parameter that will be discussed in Section 2.3. The symbols of these +Hilbert transforms are given by +(2.11) +ˆ퐻tanh +풋 +“ 푖 tanh +` +x풋, 풌yℎ +˘ +, +ˆ퐻coth +풋 +“ +# +p´푖q coth +` +x풋, 풌yℎ +˘ +, +풋 ‰ 0, +0 +풋 “ 0, +ˆ퐻sech +풋 +“ sech +` +x풋, 풌yℎ +˘ +, +ˆ퐻csch +풋 +“ +# +푖 csch +` +x풋, 풌yℎ +˘ +풋 ‰ 0, +0 +풋 “ 0. +We notice that +(2.12) +lim +ℎÑ8 +ˆ퐻tanh +풋 +“ 푖 sgn +` +x풋, 풌y +˘ +, +lim +ℎÑ8 +ˆ퐻coth +풋 +“ ´푖 sgn +` +x풋, 풌y +˘ +. +The latter coincides with the quasi-periodic Hilbert transform introduced in [73,74] in the case +of deep water while the former is its pseudo-inverse. +2.3. The quasi-periodic conformal mapping. Figure 1 illustrates a time-dependent conformal +mapping +(2.13) +푧p푤, 푡q “ 푥p훼, 훽, 푡q ` 푖푦p훼, 훽, 푡q, +푤 “ 훼 ` 푖훽, +that maps the infinite strip in the complex plane +(2.14) +푆ℎ “ t훼 ` 푖훽 : 훼 P R, ´ℎp푡q ă 훽 ă 0u +to the fluid domain +(2.15) +Ω 푓 “ tp푥, 푦q : 푥 P R, 휂b,physp푥q ă 푦 ă 휂s,physp푥, 푡qu. +To avoid ambiguity, we use 휂s,phys and 휂b,phys to denote the free surface elevation and the bot- +tom profile in physical space, respectively, whereas 휂s and 휂b are used as conformal variables +henceforth. +We assume that 푧p푤, 푡q can be extended continuously to 푆ℎ and maps the top and bottom +boundary of the strip to the free surface and the bottom boundary of the fluid domain, +respectively. Denoting +(2.16) +휁sp훼, 푡q “ 푧|훽“0p훼, 푡q “ 푥p훼, 0, 푡q ` 푖푦p훼, 0, 푡q “ 휉sp훼, 푡q ` 푖휂sp훼, 푡q, +휁bp훼, 푡q “ 푧|훽“´ℎp푡qp훼, 푡q “ 푥p훼, ´ℎp푡q, 푡q ` 푖푦p훼, ´ℎp푡q, 푡q “ 휉bp훼, 푡q ` 푖휂bp훼, 푡q, +we have +(2.17) +휂s,physp휉sp훼, 푡q, 푡q “ 휂sp훼, 푡q, +휂b,physp휉bp훼, 푡qq “ 휂bp훼, 푡q. + +QUASI-PERIODIC WATER WAVES OF FINITE DEPTH +7 +For later use in the derivation of the governing equations in conformal space, we compute the +derivative with respect to 훼 and 푡 on both sides of (2.16) and obtain that +(2.18) +푥훼 “ 휉s +훼, +푦훼 “ 휂s +훼, +푥푡 “ 휉s +푡 , +푦푡 “ 휂s +푡, +p훽 “ 0q +as well as +(2.19) +푥훼 “ 휉b +훼, +푦훼 “ 휂b +훼, +푦훼ℎ푡 ` 푥푡 “ 휉b +푡 , +´푥훼ℎ푡 ` 푦푡 “ 휂b +푡 , +p훽 “ ´ℎp푡qq +where we use the Cauchy-Riemann relation 푥훼 “ 푦훽 and 푦훼 “ ´푥훽 in the last two equalities. +The derivative of (2.17) with respect to 훼 and 푡 yields +(2.20) +휂s,phys +푥 +휉s +훼 “ 휂s +훼, +휂s,phys +푥 +휉s +푡 ` 휂s,phys +푡 +“ 휂s +푡. +and +(2.21) +휂b,phys +푥 +휉b +훼 “ 휂b +훼, +휂b,phys +푥 +휉b +푡 “ 휂b +푡 . +We are interested in the case where 휂s and 휂b are quasi-periodic functions of the form (2.5), +(2.22) +휂sp훼, 푡q “ ˜휂sp풌훼, 푡q, +˜휂sp휶, 푡q “ +ÿ +풋PZ푑 +ˆ휂s +풋p푡q푒푖x풋,휶y, +휂bp훼, 푡q “ ˜휂bp풌훼, 푡q, +˜휂bp휶, 푡q “ +ÿ +풋PZ푑 +ˆ휂b +풋 p푡q푒푖x풋,휶y, +훼 P R, 휶, 풌 P R푑, +where 풌 is fixed and its components are linearly independent over Z. This is different from +the usual conformal mapping framework [22,23,42,43,45,78], where 휂s and, if present, 휂b are +assumed to be periodic. Using the fact that 푦 is a harmonic function defined on 푆ℎ and the +boundary values of 푦 are given by 푦|훽“0 “ 휂s and 푦|훽“´ℎ “ 휂b, we obtain that +(2.23) 푦 “ 1 +ℎ +` +ˆ휂s +0 ´ ˆ휂b +0 +˘ +훽 ` ˆ휂s +0 ` +ÿ +풋‰0 +sinh +` +x풋, 풌yp훽 ` ℎq +˘ +sinh +` +x풋, 풌yℎ +˘ +ˆ휂s +풋푒푖x풋,풌y훼 ´ +ÿ +풋‰0 +sinh +` +x풋, 풌y훽 +˘ +sinh +` +x풋, 풌yℎ +˘ ˆ휂b +풋 푒푖x풋,풌y훼. +The harmonic conjugate of 푦, which is 푥, can be computed from (2.23) using the Cauchy- +Riemann equations 푥훼 “ 푦훽, 푥훽 “ ´푦훼, +(2.24) 푥 “ 1 +ℎ +` +ˆ휂s +0´ ˆ휂b +0 +˘ +훼`푥0´ +ÿ +풋‰0 +푖 +cosh +` +x풋, 풌yp훽 ` ℎq +˘ +sinh +` +x풋, 풌yℎ +˘ +ˆ휂s +풋푒푖x풋,풌y훼 ` +ÿ +풋‰0 +푖 +cosh +` +x풋, 풌y훽 +˘ +sinh +` +x풋, 풌yℎ +˘ ˆ휂b +풋 푒푖x풋,풌y훼. +Here 푥0 is an integration constant, depending on time only, that we are free to choose. Given +Ω 푓 at any time, to fix the mapping 푧, we need to specify two free parameters: ℎ and 푥0. We set +(2.25) +ℎ “ ˆ휂s +0 ´ ˆ휂b +0, +푥0 “ 0. +Hence, the first terms in (2.23) and (2.24) are just 훼 and 훽. One can choose ℎ in the same +way when the fluid domain is periodic in 푥 so that wavelengths do not change under the +conformal mapping [67]. +Alternatively, one may set ℎ “ 1, as is done in [55, 57] in the +periodic case. Setting 푥0 “ 0 requires a certain choice to be made for a parameter in the time +evolution equations [74]; this will be discussed in Section 2.5. Until then, we leave 푥0p푡q in the +representation as a time-dependent parameter. + +8 +J. WILKENING AND X. ZHAO +Comparing (2.23) and (2.24), we notice that the values of 푥 and 푦 at the top and bottom +boundary of 푆ℎ are related by the quasi-periodic Hilbert transforms of (2.10), +(2.26) +휉sp훼, 푡q “ 훼 ` 푥0p푡q ` 퐻cothr휂ssp훼, 푡q ` 퐻cschr휂bsp훼, 푡q, +휉bp훼, 푡q “ 훼 ` 푥0p푡q ´ 퐻cschr휂ssp훼, 푡q ´ 퐻cothr휂bsp훼, 푡q, +휂s +훼p훼, 푡q “ 퐻cothr휉s +훼sp훼, 푡q ` 퐻cschr휉b +훼sp훼, 푡q, +휂b +훼p훼, 푡q “ ´퐻cschr휉s +훼sp훼, 푡q ´ 퐻cothr휉b +훼sp훼, 푡q. +2.4. The quasi-periodic complex velocity potential. Let Φphysp푥, 푦, 푡q denote the velocity +potential in physical space from Section 2.1 and let 푊physp푥 ` 푖푦, 푡q “ Φphysp푥, 푦, 푡q ` +푖Ψphysp푥, 푦, 푡q be the complex velocity potential, where Ψphys is the stream function. Us- +ing the conformal mapping (2.13), we pull back these functions to the strip 푆ℎ and define +(2.27) +푊p푤, 푡q “ Φp훼, 훽, 푡q ` 푖Ψp훼, 훽, 푡q “ 푊physp푧p푤, 푡q, 푡q, +푤 “ 훼 ` 푖훽. +We denote 휑s “ Φ|훽“0, 휑b “ Φ|훽“´ℎ, 휓s “ Ψ|훽“0, 휓b “ Ψ|훽“´ℎ and use (2.16) to obtain +(2.28) +휑sp훼, 푡q “ Φphysp휉sp훼, 푡q, 휂sp훼, 푡q, 푡q “ 휑s,physp휉sp훼, 푡q, 푡q, +휑bp훼, 푡q “ Φphysp휉bp훼, 푡q, 휂bp훼, 푡q, 푡q, +휓sp훼, 푡q “ Ψphysp휉sp훼, 푡q, 휂sp훼, 푡q, 푡q “ 휓s,physp휉sp훼, 푡q, 푡q, +휓bp훼, 푡q “ Ψphysp휉bp훼, 푡q, 휂bp훼, 푡q, 푡q, +where 휑s,phys, 휓s,phys represent the values of Φphys and Ψphys on the free surface. Following [67] +for the periodic case, we assume that there is a background flow of horizontal mean velocity +U and the quasi-periodic part of 휑s has the same quasi-periods as 휂s and 휂b +(2.29) +휑sp훼, 푡q “ U훼 ` ˜휑sp풌훼, 푡q, +˜휑sp휶, 푡q “ +ÿ +풋PZ푑 +ˆ휑s +풋p푡q푒푖x풋,휶y, +훼 P R, 휶, 풌 P R푑. +According to (2.2), the bottom boundary is a streamline, therefore 휓b is a constant function +(or a function of time only). Considering that adding constants (or functions of time) to Φ and +Ψ will not affect the fluid motion, we set ˆ휑s +0 “ 0 and +(2.30) +휓b “ 0. +Since Φ and Ψ are harmonic conjugates satisfying boundary conditions (2.29) and (2.30), we +obtain +(2.31) +Φ “ U훼 ` +ÿ +풋‰0 +ˆ휑풋 +cosh +` +x풋, 풌yp훽 ` ℎq +˘ +cosh +` +x풋, 풌yℎ +˘ +푒푖x풋,풌y훼, +Ψ “ p훽 ` ℎqU ` +ÿ +풋‰0 +푖 ˆ휑풋 +sinh +` +x풋, 풌yp훽 ` ℎq +˘ +cosh +` +x풋, 풌yℎ +˘ +푒푖x풋,풌y훼. +Comparing the values of Φ and Ψ at 훽 “ 0 and 훽 “ ´ℎ, we conclude that +(2.32) +휑sp훼, 푡q “ U훼 ` 퐻cothr휓ssp훼, 푡q, +휓s +훼p훼, 푡q “ 퐻tanhr휑s +훼sp훼, 푡q, +휑b +훼p훼, 푡q “ U ` 퐻sechr휑s +훼sp훼, 푡q “ U ´ 퐻cschr휓s +훼sp훼, 푡q. + +QUASI-PERIODIC WATER WAVES OF FINITE DEPTH +9 +2.5. Governing equations in conformal space. We now present a derivation of the equations +of motion for quasi-periodic surface water waves in a conformal mapping formulation when +the fluid is of finite depth and the bottom boundary is not necessarily flat. This is an extension +of the results in [74], where the fluid depth is infinite. Since the conformal mapping is time- +dependent, even though the bottom profile in physical space is fixed, the width of the strip +in the conformal domain and the parameterization of the bottom boundary in conformal +space, denoted ℎp푡q and 휁bp훼, 푡q, respectively, both vary with time. Therefore besides the free +surface, the time evolution equations of ℎ and 휁b in conformal space are needed to describe +the evolution of the fluid domain. This is the main difference between the conformal mapping +formulations in deep and finite-depth water. +To begin, we use the chain rule to obtain +(2.33) +푑푊 +푑푤 “ 푑푊phys +푑푧 +¨ 푑푧 +푑푤 +ñ +Φphys +푥 +` 푖Ψphys +푥 +“ Φ훼 ` 푖Ψ훼 +푥훼 ` 푖푦훼 +. +Since Φphys +푦 +“ ´Ψphys +푥 +, we can express the velocity of the fluid, which is the gradient of Φphys, +in terms of Φ훼 and Ψ훼 +(2.34) +Φphys +푥 +“ Φ훼푥훼 ` Ψ훼푦훼 +푥2훼 ` 푦2훼 +, +Φphys +푦 +“ Φ훼푦훼 ´ Ψ훼푥훼 +푥2훼 ` 푦2훼 +. +Evaluating (2.34) on the free surface, we have +(2.35) +Φphys +푥 +ˇˇˇ +푧“휁sp훼,푡q “ 휑s +훼휉s +훼 ` 휓s +훼휂s +훼 +퐽s +, +Φphys +푦 +ˇˇˇ +푧“휁sp훼,푡q “ 휑s +훼휂s +훼 ´ 휓s +훼휉s +훼 +퐽s +, +퐽s “ p휉s +훼q2 ` p휂s +훼q2. +Next we derive the kinematic boundary condition in conformal space. +We define the +function +(2.36) +휗 :“ 푧푡 +푧푤 +“ 푥푡푥훼 ` 푦푡푦훼 +푥2훼 ` 푦2훼 +` 푖 푦푡푥훼 ´ 푥푡푦훼 +푥2훼 ` 푦2훼 +, +which is holomorphic on 푆ℎ as long as 푧푤 is bounded away from zero. Evaluating (2.36) at +훽 “ 0 and 훽 “ ´ℎp푡q and replacing the derivatives of 푥 and 푦 by the derivatives of 휉s and 휂s +using (2.18), (2.19), we obtain that +(2.37) +Re 휗 +ˇˇˇ +훽“0 “ +휉s +푡휉s +훼 ` 휂s +푡휂s +훼 +퐽s +, +Im 휗 +ˇˇˇ +훽“0 “ +휂s +푡휉s +훼 ´ 휉s +푡휂s +훼 +퐽s +, +(2.38) +Re 휗 +ˇˇˇ +훽“´ℎp푡q “ +휉b +푡 휉b +훼 ` 휂b +푡 휂b +훼 +퐽b +, +Im 휗 +ˇˇˇ +훽“´ℎp푡q “ +휂b +푡 휉b +훼 ´ 휉b +푡 휂b +훼 +퐽b +` ℎ푡, +퐽b “ p휉b +훼q2 ` p휂b +훼q2. +Furthermore, the substitution of (2.20) and (2.35) into (2.3) gives +(2.39) +휂s +푡휉s +훼 ´ 휉s +푡휂s +훼 “ ´휓s +훼. +Therefore we have +(2.40) +Im 휗 +ˇˇˇ +훽“0 “ ´휓s +훼 +퐽s . + +10 +J. WILKENING AND X. ZHAO +Substituting (2.21) into (2.38), we obtain that 휂b +푡 휉b +훼 ´ 휉b +푡 휂b +훼 “ 0, thus +(2.41) +Im 휗 +ˇˇˇ +훽“´ℎp푡q “ ℎ푡. +Since ℎ푡 does not depend on the spatial variable, similar to (2.32), Re 휗|훽“0 and Re 휗|훽“´ℎp푡q +can be determined by Im 휗|훽“0 up to an additive constant (that may depend on time but not +space) as follows, +(2.42) +휉s +푡휉s +훼 ` 휂s +푡휂s +훼 +퐽s +“ ´퐻coth +„휓s +훼 +퐽s +ȷ +` 퐶1, +휉b +푡 휉b +훼 ` 휂b +푡 휂b +훼 +퐽b +“ 퐻csch +„휓s +훼 +퐽s +ȷ +` 퐶1. +Since 휗 is a holomorphic function defined on 푆ℎ, using Cauchy’s integral theorem, we obtain +(2.43) +ż 푎`푖p휖´ℎq +´푎`푖p휖´ℎq +` +ż 푎´푖휖 +푎`푖p휖´ℎq +` +ż ´푎´푖휖 +푎´푖휖 +` +ż ´푎`푖p휖´ℎq +´푎´푖휖 +휗p푤q 푑푤 “ 0, +푎, 휖 ą 0. +Dividing both sides of (2.43) by 2푎 and taking the limit 푎 Ñ 8, 휖 Ñ 0`, we have +(2.44) +ˆ휗0 “ 푃0r휗p훼qs “ 푃0r휗p훼 ´ 푖ℎqs, +where we use (2.7) in the first equality. Substituting (2.40) and (2.41) into (2.44), we obtain the +the time evolution equation of the width of the strip 푆ℎ +(2.45) +ℎ푡 “ ´푃0 +„휓s +훼 +퐽s +ȷ +. +Finally, combining (2.37), (2.38) and (2.42), we obtain the kinematic boundary conditions at +both the free surface and the bottom boundary in conformal space +(2.46) +˜휉s +푡 +휂s +푡 +¸ +“ +˜휉s +훼 +´휂s +훼 +휂s +훼 +휉s +훼 +¸ ¨ +˝´퐻coth ”휓s +훼 +퐽s +ı +` 퐶1 +´휓s +훼 +퐽s +˛ +‚, +˜휉b +푡 +휂b +푡 +¸ +“ +˜휉b +훼 +휂b +훼 +¸ ˆ +퐻csch +„휓s +훼 +퐽s +ȷ +` 퐶1 +˙ +. +Since 휉s and 휉b are determined by 휂s and 휂b up to an additive constant 푥0 by (2.26), we only +need to evolve ℎ, 휂s and 휂b to track the evolution of the fluid domain. Comparing (2.24) and +(2.46), we know that the free parameter 푥0 is related to 퐶1 through the ODE +(2.47) +푑푥0 +푑푡 “ 푃0 +„ +휉s +훼 +ˆ +´퐻coth +„휓s +훼 +퐽s +ȷ +` 퐶1 +˙ +` 휂s +훼휓s +훼 +퐽s +ȷ +. +Thus, 푥0p푡q is uniquely determined by 퐶1 and 푥0p0q. Several choices of 퐶1 have been discussed +in detail in [74]. In the scope of this paper, we choose 퐶1 and 푥0p0q as follows +(2.48) +퐶1 “ 푃0 +“ +휉s +훼퐻cothr휓s +훼{퐽ss ´ 휂s +훼휓s +훼{퐽s‰ +, +푥0p0q “ 0. +This ensures that +(2.49) +푥0p푡q “ 0, +p푡 ě 0q +and alleviates the need to explicitly solve the ODE (2.47). +Now we derive the dynamic boundary condition at the free surface in conformal space from +(2.4). Differentiating the first equation in (2.28) with respect to 푡, we obtain +(2.50) +휑s +푡 “ 휑s,phys +푥 +휉s +푡 ` 휑s,phys +푡 +, +where 휑s,phys +푥 +can be expressed in terms of the gradient of Φphys as follows +(2.51) +휑s,phys +푥 +“ Φphys +푥 +` Φphys +푦 +휂s,phys +푥 +. + +QUASI-PERIODIC WATER WAVES OF FINITE DEPTH +11 +From (2.33), we know that +ˇˇ∇Φphysˇˇ2 “ +´` +휑s +훼 +˘2 ` +` +휓s +훼 +˘2¯ +{퐽s at 훽 “ 0. Substitution of (2.46), +(2.50) and (2.51) into (2.4) then gives +(2.52) +휑s +푡 “ +` +Φphys +푥 +, Φphys +푦 +˘ ˆ휉s +훼 +´휂s +훼 +휂s +훼 +휉s +훼 +˙ +looooooooooooooooomooooooooooooooooon +` +휑s +훼 , ´휓s +훼 +˘ +ˆ´퐻coth r휓s +훼{퐽ss ` 퐶1 +´휓s +훼{퐽s +˙ +´ +` +휑s +훼 +˘2 ` +` +휓s +훼 +˘2 +2퐽s +´ 푔휂s ` 휏휅 ` 퐶, +where 휅 is the mean curvature, given by +(2.53) +휅 “ 휉s +훼휂s +훼훼 ´ 휂s +훼휉s +훼훼 +` +퐽s˘3{2 +, +and 퐶 is an arbitrary integration constant that may depend on time but not space. In the +discussion of this paper, we choose 퐶 such that 푃0r휑s +푡s “ 0. +In conclusion, (2.45), (2.46) +and (2.52) are the governing equations in conformal space for finite-depth quasi-periodic +gravity-capillary waves. +Following [74], instead of solving these equations directly, which are posed on the real line, +we lift the problem to a higher dimensional torus T푑 and compute the time evolution of the +corresponding torus functions; then we evaluate the torus functions along the characteristic +direction to obtain quasi-periodic functions on the real line. Using the torus version of the +quasi-periodic derivative and Hilbert transform operators in Definitions 2.2 and 2.3, we obtain +the governing equations on T푑 from (2.45), (2.46) and (2.52), +(2.54) +˜휂s +푡 “ +` +´ 퐻cothr ˜휒ss ` 퐶1 +˘ +˜휂s +훼 ´ ˜휉s +훼 ˜휒s, +˜휂b +푡 “ +` +퐻cschr ˜휒ss ` 퐶1 +˘ +˜휂b +훼, +ℎ푡 “ ´푃0r ˜휒ss, +˜휑s +푡 “ 푃 +„` ˜휓s +훼 +˘2 ´ +` +˜휑s +훼 ` U +˘2 +2˜퐽s +` +` +퐶1 ´ 퐻cothr ˜휒ss +˘` +˜휑s +훼 ` U +˘ +´ 푔 ˜휂s ` 휏˜휅 +ȷ +˜휉s “ 퐻cothr˜휂ss ` 퐻cschr˜휂bs, +˜휓s “ 퐻tanhr ˜휑ss, +˜퐽s “ +` +1 ` ˜휉s +훼 +˘2 ` +` +˜휂s +훼 +˘2, +˜휒s “ +˜휓s +훼 +˜퐽s , +˜휅 “ +` +1 ` ˜휉s +훼 +˘ +˜휂s +훼훼 ´ ˜휂s +훼 ˜휉s +훼훼 +p˜퐽sq3{2 +, +퐶1 “ 푃0 +“` +1 ` ˜휉s +훼 +˘ +퐻cothr ˜휒ss ´ ˜휂s +훼 ˜휒s‰ +. +We remark that ˜휑, which is defined on T푑, represents only the quasi-periodic part of 휑. An +extra term U훼 is included in the definition (2.29) to account for the background flow (when +present). Similarly, 휉s and 휉b are obtained from ˜휉s and ˜휉b via +(2.55) +휉sp훼, 푡q “ 훼 ` ˜휉sp풌훼, 푡q, +휉bp훼, 푡q “ 훼 ` ˜휉bp풌훼, 푡q, +where ˜휉s is given in (2.54) and ˜휉b is given by +(2.56) +˜휉b “ ´퐻cschr˜휂ss ´ 퐻cothr˜휂bs. +According to (2.55), we have +(2.57) +휉s +훼p훼, 푡q “ 1 ` ˜휉s +훼p풌훼, 푡q, +휉b +훼p훼, 푡q “ 1 ` ˜휉b +훼p풌훼, 푡q, +which is the reason p1 ` ˜휉s +훼q appears in various places in (2.54). + +12 +J. WILKENING AND X. ZHAO +Remark 2.4. Modifying the analysis of [74], one can show that if 휁s and 휁b are injective, then +휂s,phys and 휂b,phys are also quasi-periodic functions of the same quasi-periods. Moreover, the +corresponding torus functions ˜휂s,phys and ˜휂b,phys can be obtained from ˜휂s and ˜휂b by +(2.58) +˜휂s,physp풙, 푡q “ ˜휂sp풙 ` 풌 ˜Asp풙, 푡q, 푡q, +˜휂b,physp풙, 푡q “ ˜휂bp풙 ` 풌 ˜Abp풙, 푡q, 푡q, +where ˜As and ˜Ab satisfy +(2.59) +˜Asp풙, 푡q ` ˜휉sp풙 ` 풌 ˜Asp풙, 푡q, 푡q “ 0, +˜Abp풙, 푡q ` ˜휉bp풙 ` 풌 ˜Abp풙, 푡q, 푡q “ 0. +In numerical computations, at any given 푡, one can formulate (2.58) as a nonlinear least- +squares problem and solve it using a Leveberg-Marquardt method [72], which is discussed in +Appendix A. +Remark 2.5. Since the bottom boundary is stationary, conservation of mass requires that the +mean surface height, which we denote by +(2.60) +휇 “ +1 +p2휋q푑 +ż +T푑 ˜휂phys 푑푥1 ¨ ¨ ¨ 푑푥푑 “ +1 +p2휋q푑 +ż +T푑 ˜휂p1 ` ˜휉훼q 푑훼1 ¨ ¨ ¨ 푑훼푑, +is a constant in time. Indeed, one finds that 휇푡 “ 0 by differentiating the second formula of +(2.60) under the integral sign, integrating the term ˜휂 ˜휉훼푡 by parts with respect to 훼, and using +(2.39). One usually assumes 휇 “ 0, though for traveling waves it is convenient to first compute +the wave assuming ˆ휂0 “ 0 and then adjust ˆ휂0 at the end to achieve 휇 “ 0. +Remark 2.6. The governing equations (2.54) still hold when the bottom boundary of the fluid +domain is flat: 휂b,physp푥q “ ´ℎphys. In the usual case that 휇 “ 0, ℎphys is the mean depth of +the fluid in physical space. Otherwise the mean depth is 휇 ` ℎphys. From (2.25), we have +(2.61) +휂b “ ´ℎphys “ ˆ휂s +0 ´ ℎ, +which is a constant independent of 훼 and 푡 even though ℎ and ˆ휂s +0 vary in time. Moreover, 휉s +is related to 휂s by +(2.62) +휉s “ 훼 ` 퐻cothr휂ss. +Therefore, when the bottom boundary is flat, one only needs to evolve ˜휂s, ˜휑s and ℎ. +Remark 2.7. Even though we derive (2.54) in the quasi-periodic setting, these equations +still hold for the periodic problem if we set 푑 “ 1 and 풌 “ p1q. To obtain the governing +equations on T, one just needs to replace the quasi-periodic Hilbert transforms by their +periodic counterparts in (2.54): +(2.63) +퐻tanhr 푓 sp훼q “ +ÿ +푗‰0 +푖 tanhp푗ℎq ˆ푓푗푒푖푗훼, +퐻cothr 푓 sp훼q “ +ÿ +푗‰0 +p´푖q cothp푗ℎq ˆ푓푗푒푖푗훼, +퐻cschr 푓 sp훼q “ +ÿ +푗‰0 +푖 cschp푗ℎq ˆ푓푗푒푖푗훼, +퐻sechr 푓 sp훼q “ +ÿ +푗‰0 +sechp푗ℎq ˆ푓푗푒푖푗훼, +where 푓 is defined on T. If 푑 ą 1, the periodic problem may be embedded in the quasi-periodic +problem by assuming that each of the torus functions in (2.54) is independent of 훼2, . . . , 훼푑. +We conclude this section by explaining how to compute the trajectories of fluid particles in +the conformal mapping formulation. We denote the trajectory of a fluid particle by +(2.64) +푧푝p푡q “ 푥푝p푡q ` 푖푦푝p푡q “ 푥p훼푝p푡q, 훽푝p푡q, 푡q ` 푖푦p훼푝p푡q, 훽푝p푡q, 푡q. +Unlike physical space, where the particle trajectory is computed through +(2.65) +9푥푝 “ Φphys +푥 +p푥푝p푡q, 푦푝p푡q, 푡q, +9푦푝 “ Φphys +푦 +p푥푝p푡q, 푦푝p푡q, 푡q, + +QUASI-PERIODIC WATER WAVES OF FINITE DEPTH +13 +in conformal space, one needs to compute the time evolution of 훼푝p푡q and 훽푝p푡q since the +conformal mapping is time-dependent. To do so, we differentiate (2.64) and use (2.65) to +obtain +(2.66) +9푧푝 “ Φphys +푥 +` 푖Φphys +푦 +“ +` +푥훼 9훼푝 ´ 푦훼 9훽푝 ` 푥푡 +˘ +` 푖 +` +푦훼 9훼푝 ` 푥훼 9훽푝 ` 푦푡 +˘ +, +where we use the fact that 푥 and 푦 are harmonic conjugates in the last equality. Therefore we +have +(2.67) +˜ +9훼푝 +9훽푝 +¸ +“ +1 +푥2훼 ` 푦2훼 +˜ +푥훼 +푦훼 +´푦훼 +푥훼 +¸ ˜ +Φphys +푥 +´ 푥푡 +Φphys +푦 +´ 푦푡 +¸ +. +Substituting (2.34) and (2.36) into (2.67), we obtain the time evolution equation of 훼푝 and 훽푝 +as follows +(2.68) +˜ +9훼푝 +9훽푝 +¸ +“ +1 +푥2훼 ` 푦2훼 +˜ +Φ훼 +´Ψ훼 +¸ +´ +˜ +Rep푧푡{푧푤q +Imp푧푡{푧푤q +¸ +. +On the free surface, according to (2.40) and (2.42), (2.68) reads +(2.69) +˜ +9훼푝 +9훽푝 +¸ +“ 1 +퐽푠 +˜ +휑s +훼 +´휓s +훼 +¸ +` +˜ +퐻cothr휓s +훼{퐽푠s ´ 퐶1 +휓s +훼{퐽푠 +¸ +“ +˜ +휑s +훼{퐽푠 ` 퐻cothr휓s +훼{퐽푠s ´ 퐶1 +0 +¸ +. +Thus, 훽푝p푡q “ 훽푝p0q “ 0 and we only need to evolve 훼푝 to track the particle trajectory on the +free surface. +3. Quasi-periodic traveling waves +3.1. Governing equations of quasi-periodic traveling waves. For traveling waves, the system +should be translation invariant, so we assume the bottom boundary is flat. According to +Remark 2.6, we only need to consider the surface variables in this case. Thus, to simplify the +notation, we drop the superscript “s” in these variables in this section. Moreover, as discussed +in Section 2.1, we focus our discussion on the laboratory frame and assume that there is no +background flow. +Since the bottom boundary is flat, 휉, 휂 and 휑, 휓 are related by Hilbert transforms +(3.1) +휉 “ 훼 ` 퐻cothr휂s, +휉훼 “ 1 ` 퐻cothr휂훼s, +휑 “ 퐻cothr휓s, +휑훼 “ 퐻cothr휓훼s. +We assume the wave is traveling from left to right at speed 푐; therefore, we have +(3.2) +휂physp푥, 푡q “ 휂phys +0 +p푥 ´ 푐푡q, +휑physp푥, 푡q “ 휑phys +0 +p푥 ´ 푐푡q. +Differentiating both sides of (3.2) with respect to 푥 and 푡 separately, we know that a traveling +solution satisfies +(3.3) +휂phys +푡 +“ ´푐휂phys +푥 +, +휑phys +푡 +“ ´푐휑phys +푥 +. +Substituting the second equation of (2.20) into the first equation of (3.3) and multiplying both +sides of the equation by 휉s +훼, we obtain +(3.4) +휂푡휉훼 ´ 휉푡휂훼 “ ´푐휂훼. +Comparing (3.4) and (2.39), we conclude that a traveling solution satisfies +(3.5) +휓훼 “ 푐휂훼 +in conformal space. Applying the Hilbert transform 퐻coth to both sides of (3.5), we obtain +(3.6) +휑훼 “ 푐p휉훼 ´ 1q. + +14 +J. WILKENING AND X. ZHAO +Substituting the traveling condition of 휑phys in (3.3) into (2.50) and employing (2.51) to express +휑phys +푥 +in terms of the gradient of Φphys, we obtain that +(3.7) +휑푡 “ +` +Φphys +푥 +` Φphys +푦 +휂phys +푥 +˘ +p휉푡 ´ 푐q +“ 휑훼 +휉훼 +` +휉훼 +` +´ 퐻coth”휓훼 +퐽 +ı +` 퐶1 +˘ +` 휂훼 +휓훼 +퐽 +´ 푐 +˘ +“ 휑훼 +휉훼 +ˆ +휉훼 +` +´ 퐻coth”휓훼 +퐽 +ı +` 퐶1 +˘ +` +푐 +` +휂훼 +˘2 +퐽 +´ 푐 +˙ +“ 휑훼 +ˆ +´ 퐻coth”휓훼 +퐽 +ı +` 퐶1 ´ 푐휉훼 +퐽 +˙ +. +Here in the second equality, we use the first equation in (2.20) to rewrite 휂phys +푥 +as 휂s +훼{휉s +훼 and +substitute the gradient of Φphys and 휉푡 using (2.35) and (2.46), respectively. In the third equality, +we use (3.5) to replace 휓훼 by 푐휂훼. The substitution of (3.5) and (3.7) into (2.52) gives +(3.8) +푐 +퐽 +` +휑훼휉훼 ` 휓훼휂훼 +˘ +´ 1 +2퐽 +` +p휑훼q2 ` p휓훼q2˘ +´ 푔휂 ` 휏휅 ` 퐶 “ 0. +Using (3.5) and (3.6) to express 휑훼 and 휓훼 in terms of 휉훼 and 휂훼, respectively, we obtain the +governing equation of traveling waves +(3.9) +푃 +„ 푐2 +2퐽 ` 푔휂 ´ 휏휅 +ȷ +“ 0, +where we choose the integration constant 퐶 in (3.8) such that 푃0 acting on the left-hand side of +(3.8) returns zero. Since (3.9) does not depend on time, the solution of (3.9) can be considered +as the initial condition of a traveling wave. From (3.1) and (2.53), we know that 퐽 and 휅 are +determined by 휂; hence, the unknowns in (3.9) are 휏, 푐 and 휂. Even though we are mainly +interested in the case where 휂 is quasi-periodic, the governing equation (3.9) still holds when +휂 is periodic. Due to the projection operator, modifying 휂 by a constant will not influence +(3.9); hence, we assume that 푃0r휂s “ 0. In this paper, we focus on traveling waves with even +symmetry +(3.10) +휂p훼q “ 휂p´훼q. +We compute 휉 from 휂 using (3.1) and deduce that 휉 is odd. Asymmetric traveling waves have +been studied in [31,68,79] in the periodic setting. +As in the initial value problem, we first solve for ˜휂 on T푑 and then reconstruct 휂 from ˜휂 +using (2.22). The governing equations of traveling waves on the torus read +(3.11) +Rr휏, 푏, ˜휂s “ 푃 +„ 푏 +2˜퐽 +` 푔 ˜휂 ´ 휏˜휅 +ȷ +“ 0, +˜휉 “ 퐻cothr˜휂s, +˜퐽 “ +` +1 ` ˜휉훼 +˘2 ` ˜휂2 +훼, +˜휅 “ +` +1 ` ˜휉훼 +˘ +˜휂훼훼 ´ ˜휂훼 ˜휉훼훼 +˜퐽3{2 +, +where 푏 “ 푐2 and R is called the residual function. We treat the strip width ℎ in conformal +space as a fixed parameter and suppress it in the argument list of R; see Remark 3.1 below. +Linearizing (3.11) around the zero solution ˜휂 “ 0, we obtain +(3.12) +푏퐻cothr훿 ˜휂훼s ´ 푔훿 ˜휂 ` 휏훿 ˜휂훼훼 “ 0, + +QUASI-PERIODIC WATER WAVES OF FINITE DEPTH +15 +where 훿 ˜휂 denotes the variation of ˜휂. Expressing 훿 ˜휂 in terms of its Fourier series in (3.12), we +obtain the dispersion relation for the linearized problem +(3.13) +푏 cothpx풋, 풌yℎqx풋, 풌y ´ 푔 ´ 휏px풋, 풌yq2 “ 0, +풋 P 푍푑. +Since the entries of 풌 are linearly independent over Z, given 푏 and 휏, there exist at most two +linearly independent vectors 풋1, 풋2 P Z푑 that satisfy the dispersion relation [14]. For simplicity, +we consider the basic case where 푑 “ 2; hence, 휂 possesses two quasi-periods and ˜휂 is defined +on T2. Without loss of generality, we also assume that 풋1 “ p1, 0q푇, 풋2 “ p0, 1q푇 and 풌 “ p1, 푘q푇, +where 푘 is a positive irrational number. +In summary, we study quasi-periodic traveling waves of the following form +(3.14) +휂p훼q “ ˜휂p훼, 푘훼q, +˜휂p훼1, 훼2q “ +ÿ +푗1,푗2PZ +ˆ휂푗1,푗2푒푖p푗1훼1`푗2훼2q. +We also assume that ˜휂 is an even function with zero mean on T2, which is consistent with the +assumptions on 휂. Therefore the Fourier coefficients of ˜휂 satisfy +(3.15) +ˆ휂0,0 “ 0, +ˆ휂푗1,푗2 “ ˆ휂´푗1,´푗2 P R. +Here ˜휂 has zero mean in conformal space. We refer to Remark 3.1 below if one wants to obtain +solutions with zero mean in physical space. Under assumptions (3.14) and (3.15), we can study +the problem of quasi-periodic traveling waves in the setting of a bifurcation problem with a +two-dimensional kernel spanned by the solutions of the linearized problem (3.12): +(3.16) +˜휂linp훼1, 훼2q “ ˆ휂1,0p푒푖훼1 ` 푒´푖훼1q ` ˆ휂0,1p푒푖훼2 ` 푒´푖훼2q, +푏lin “ 푐2 +lin “ +푔p푘2 ´ 1q +푘p푘 cothpℎq ´ cothp푘ℎqq, +휏lin “ 푔p푘 cothp푘ℎq ´ cothpℎqq +푘p푘 cothpℎq ´ cothp푘ℎqq . +We refer to ˆ휂1,0 and ˆ휂0,1 as the base Fourier coefficients and the corresponding Fourier modes +푒˘푖훼1, 푒˘푖훼2 as the base Fourier modes. Nonlinear solutions can be considered as bifurcations +from the zero-amplitude solution. We usually choose the base Fourier coefficients as bifur- +cation parameters and fix them at nonzero values to ensure that the solutions we obtain are +genuinely quasi-periodic. In finite depth, ℎ is a third parameter. +As shown in [75], large-amplitude quasi-periodic traveling solutions can often be found +by searching for secondary bifurcations from finite-amplitude periodic traveling waves. The +linearization of (3.11) around a periodic solution reads +(3.17) +훿R “ 푃 +« +훿푏 +2˜퐽 +´ 1 +2˜퐽2 푏훿˜퐽 ` 푔훿 ˜휂 ´ 훿휏˜휅 ´ 휏훿 ˜휅 +ff +, +훿 ˜휉훼 “ 퐻cothr훿 ˜휂훼s, +훿˜퐽 “ 2 +´ +p1 ` ˜휉훼q훿 ˜휉훼 ` ˜휂훼훿 ˜휂훼 +¯ +, +훿 ˜휅 “ ´3 +2 ˜휅 훿˜퐽 +˜퐽 +` +1 +˜퐽3{2 +´ +훿 ˜휉훼 ˜휂훼훼 ` p1 ` ˜휉훼q훿 ˜휂훼훼 ´ 훿 ˜휂훼 ˜휉훼훼 ´ ˜휂훼훿 ˜휉훼훼 +¯ +. +Let 푞 denote the triple p휏, 푏, ˜휂q and let 픮perp푠q denote a one-parameter family of periodic trav- +eling waves embedded in the quasi-periodic framework by assuming ˜휂p훼1, 훼2q is independent +of 훼2. Here 푠 is an amplitude parameter (such as ˆ휂1,0), and, for simplicity, we fix 휏 and the strip +width ℎ in conformal space to be independent of 푠. Each solution 푞 “ 픮perp푠q in the family +satisfies R +` +푞 +˘ +“ 0. In [75], an algorithm is presented for locating bifurcation points by using a + +16 +J. WILKENING AND X. ZHAO +quadratically convergent root bracketing technique [13] to locate zeros of the signed smallest +singular value +(3.18) +휒p푠q “ sgn +´ +det +´ +J quap푠q +¯¯ +휎min +´ +J quap푠q +¯ +. +Here J quap푠q is a Fourier truncation of the restricted Jacobian obtained from the linearization +(3.17) applied only in quasi-periodic perturbation directions of the form 훿푞 “ p0, 0, 훿 ˜휂quaq, +where 훿 ˜휂qua has 2D Fourier modes x +훿 ˜휂qua +푗1,푗2 that are all zero unless 푗2 P t1, ´1u. This construction +is based on Bloch-Fourier perturbation theory over periodic potentials [41]. At zeros of 휒p푠q, +J quap푠q has a kernel that provides a bifurcation direction 훿 ˜휂qua that allows us to switch +from the primary periodic branch to the secondary quasi-periodic branch of traveling waves. +We use ˜휂per ` 휖훿 ˜휂qua, with 휖 chosen empirically, as an initial guess for solutions on this +secondary branch, and then use numerical continuation to follow the branch beyond the +realm of linearization about the primary branch. +Further discussion of the analysis and +computation of the bifurcation problem in the infinite-depth setting is given in [75]. +Remark 3.1. We have simplified the computation of quasi-periodic traveling waves via the +conformal mapping formulation by setting ˆ휂0,0 “ 0 and fixing the strip width ℎ in conformal +space. The mean surface height and depth of the bottom boundary in physical space, 휇 and +ℎphys, can then be computed from (2.60) and (2.61), respectively. If desired, after computing +a solution with ˆ휂0,0 “ 0, one can adjust the height of the traveling wave by setting ˆ휂0,0 “ ´휇 +and 휇 “ 0, assigned in that order. The resulting solution satisfies +(3.19) +ˆ휂0,0 “ ´푃0r +` +푃r˜휂s +˘ +p1 ` ˜휉훼qs, +where 푃r˜휂s on the right-hand side is the initially computed solution with ˆ휂0,0 “ 0. Another +option is to prescribe 휇 “ 0, ℎphys, ˆ휂1,0 and ˆ휂0,1 and solve for ˆ휂0,0 and ℎ along with the +remaining Fourier modes ˆ휂푗1,푗2 using the Levenberg-Marquardt solver. +This would entail +including ℎ “ ℎphys ` ˆ휂0,0 from (2.61) as well as (3.19) as additional constraints in (3.11). +3.2. Weakly nonlinear approximations of quasi-periodic traveling waves. Although the +primary focus of this work is on computing quasi-periodic solutions of the fully nonlinear +time-dependent and traveling water wave equations in finite depth, it is instructive to in- +vestigate how small divisors arise in weakly nonlinear approximations of small-amplitude +quasi-periodic traveling waves. In previous work, it has been necessary to treat such small +divisors carefully using Nash-Moser theory [37, 53] to prove existence of temporally quasi- +periodic water waves [7–10,27]. Here we focus on spatial quasi-periodicity. +As discussed in Section 3.1, the traveling solutions bifurcating from the zero solution form +a three-parameter family with bifurcation parameters ˆ휂1,0, ˆ휂0,1 and ℎ. In the weakly nonlinear +model, we treat ℎ as a constant and set these two Fourier coefficients to be fixed, non-zero +multiples of an amplitude parameter 휖 and aim to express 푏, 휏 and the other Fourier coefficients +of ˜휂 in terms of them. Let us consider the following asymptotic expansions of 푏, 휏 and ˜휂 +(3.20) +푏 “푏p0q ` 휖푏p1q ` 휖2푏p2q ` 휖3푏p3q ` 푂p휖4q, +휏 “휏p0q ` 휖휏p1q ` 휖2휏p2q ` 휖3휏p3q ` 푂p휖4q, +˜휂 “휖 ˜휂p1q ` 휖2 ˜휂p2q ` 휖3 ˜휂p3q ` 푂p휖4q. + +QUASI-PERIODIC WATER WAVES OF FINITE DEPTH +17 +Substituting (3.20) into (3.11) and eliminating the coefficients of 휖푛 for 푛 “ 0, 1, 2, we obtain +(3.21) +푂p1q : +푃 +„1 +2푏p0q +ȷ +“ 0, +푂p휖q : +푃 +„1 +2푏p1q ` 푔 ˜휂p1q ´ 푏p0q퐻coth“ +˜휂p1q +훼 +‰ +´ 휏p0q ˜휂p1q +훼훼 +ȷ +“ 0, +푂p휖2q : +푃 +„1 +2푏p2q ` 푔 ˜휂p2q ´ 푏p0q퐻coth“ +˜휂p2q +훼 +‰ +´ 휏p0q ˜휂p2q +훼훼 +´ 푏p1q퐻coth“ +˜휂p1q +훼 +‰ +´ 휏p1q ˜휂p1q +훼훼 +` 푏p0q +ˆ3 +2 +´ +퐻coth“ +˜휂p1q +훼 +‰¯2 +´ 1 +2 +` +˜휂p1q +훼 +˘2 +˙ +` 휏p0q ´ +2퐻coth“ +˜휂p1q +훼 +‰ +˜휂p1q +훼훼 ` 퐻coth“ +˜휂p1q +훼훼 +‰ +˜휂p1q +훼 +¯ ȷ +“ 0. +Since the constant term in (3.21) vanishes under the projection, we rewrite the second equation +as +(3.22) +푃 +„ +푏p0q퐻coth“ +˜휂p1q +훼 +‰ +´ 푔 ˜휂p1q ` 휏p0q ˜휂p1q +훼훼 +ȷ +“ 0, +which is the same as the linearization (3.12); therefore, we have +(3.23) +˜휂p1q “ ˜휂lin “ ˆ휂1,0푒푖훼1 ` ˆ휂0,1푒푖훼2 ` 푐.푐., +푏p0q “ 푏lin, +휏p0q “ 휏lin. +Using the property of the projection operator and the assumption that 푃0r˜휂s “ 0, we rewrite +the third equation in (3.21) as +(3.24) +푔 ˜휂p2q ´ 푏p0q퐻coth“ +˜휂p2q +훼 +‰ +´ 휏p0q ˜휂p2q +훼훼 +loooooooooooooooooooomoooooooooooooooooooon +퐴p2q +´푏p1q퐻coth“ +˜휂p1q +훼 +‰ +´ 휏p1q ˜휂p1q +훼훼 +loooooooooooooooomoooooooooooooooon +퐵p2q +“ 푃 +„ +푏p0q +ˆ +´3 +2 +´ +퐻coth“ +˜휂p1q +훼 +‰¯2 +` 1 +2 +` +˜휂p1q +훼 +˘2 +˙ +´ 휏p0q ´ +2퐻coth“ +˜휂p1q +훼 +‰ +˜휂p1q +훼훼 ` 퐻coth“ +˜휂p1q +훼훼 +‰ +˜휂p1q +훼 +¯ȷ +looooooooooooooooooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooooooooooooooooon +퐶p2q +. +Substituting ˜휂p1q, ˜푏p0q and ˜휏p0q into 퐶p2q using (3.23), we obtain +(3.25) +퐶p2q “ ˆ퐶p2q +2,0푒푖p2훼1q ` ˆ퐶p2q +0,2푒푖p2훼2q ` ˆ퐶p2q +1,1푒푖p훼1`훼2q ` ˆ퐶p2q +1,´1푒푖p훼1´훼2q ` 푐.푐., +where the Fourier coefficients of 퐶p2q are +(3.26) +ˆ퐶p2q +2,0 “ 푔 ˆ휂2 +1,0 +3p푘2 ` 1q coth2pℎq ´ 6푘 cothp푘ℎq cothpℎq ` 푘2 ´ 1 +2푘pcothp푘ℎq ´ 푘 cothpℎqq +, +ˆ퐶p2q +0,2 “ ´푔푘 ˆ휂2 +0,1 +3p푘2 ` 1q coth2p푘ℎq ´ 6푘 cothp푘ℎq cothpℎq ´ 푘2 ` 1 +2pcothp푘ℎq ´ 푘 cothpℎqq +, +ˆ퐶p2q +1,1 “ ´푔 ˆ휂1,0 ˆ휂0,1 +p푘2 ` 2푘q coth2p푘ℎq ´ p2푘 ` 1q coth2pℎq ` p´푘2 ` 1q cothp푘ℎq cothpℎq ´ 푘2 ` 1 +cothp푘ℎq ´ 푘 cothpℎq +, +ˆ퐶p2q +1,´1 “ 푔 ˆ휂1,0 ˆ휂0,1 +p푘2 ´ 2푘q coth2p푘ℎq ` p2푘 ´ 1q coth2pℎq ` p푘2 ´ 1q cothp푘ℎq cothpℎq ´ 푘2 ` 1 +cothp푘ℎq ´ 푘 cothpℎq +. + +18 +J. WILKENING AND X. ZHAO +We observe that 퐴p2q is linear with respect to ˜휂p2q and the Fourier coefficients of 퐴p2q can be +expressed as +(3.27) +ˆ퐴p2q +푗1,푗2 “ ˆ푆푗1,푗2 ˆ휂p2q +푗1,푗2, +where the symbol ˆ푆푗1,푗2 is defined by +(3.28) +ˆ푆푗1,푗2 “ 푔 ´ 푏p0q cothpp푗1 ` 푘푗2qℎqp푗1 ` 푘푗2q ` 휏p0qp푗1 ` 푘푗2q2 +“ 푔 +푘 +ˆ +푘 ` +푘2 ´ 1 +cothp푘ℎq ´ 푘 cothpℎq cothpp푗1 ` 푘푗2qℎqp푗1 ` 푘푗2q ` cothpℎq ´ 푘 cothp푘ℎq +cothp푘ℎq ´ 푘 cothpℎqp푗1 ` 푘푗2q2 +˙ +. +Since ˆ푆˘1,0 and ˆ푆0,˘1 are both zero according to the definition, we know that ˆ퐴p2q +˘1,0 “ ˆ퐴p2q +0,˘1 “ 0. +We also observe that 퐵p2q is linear with respect to ˜휂p1q with Fourier coefficients +(3.29) +ˆ퐵p2q +푗1,푗2 “ ˆ푄p1q +푗1,푗2 ˆ휂p1q +푗1,푗2, +ˆ푄p푛q +푗1,푗2 “ ´푏p푛q cothpp푗1 ` 푘푗2qℎqp푗1 ` 푘푗2q ` 휏p푛qp푗1 ` 푘푗2q2, +where p푗1, 푗2q “ p˘1, 0q, p0, ˘1q according to (3.23). Combining (3.26), (3.27) and (3.29), we +obtain +(3.30) +푏p1q “ 휏p1q “ 0, +ˆ휂p2q +푗1,푗2 “ +$ +’ +& +’ +% +퐶p2q +푗1,푗2 +ˆ푆푗1,푗2 +, +|푗1| ` |푗2| “ 2, +0, +|푗1| ` |푗2| ‰ 2. +Remark 3.2. One can obtain the asymptotic expansions of quasi-periodic traveling waves in +the case of deep water by letting ℎ go to infinity. In this case, the expressions of ˜휂p1q, 푏p0q and +휏p0q read +(3.31) +˜휂p1q “ ˆ휂1,0푒푖훼1 ` ˆ휂0,1푒푖훼2 ` 푐.푐., +푏p0q “ 푔 ` 푔 +푘 , +휏p0q “ 푔 +푘 +and the expressions of ˜휂p2q, 푏p1q and 휏p1q read +(3.32) +˜휂p2q “ ˆ휂p2q +2,0푒푖p2훼1q ` ˆ휂p2q +0,2푒푖p2훼2q ` ˆ휂p2q +1,1푒푖p훼1`훼2q ` ˆ휂p2q +1,´1푒푖p훼1´훼2q ` 푐.푐., +푏p1q “ 휏p1q “ 0, +ˆ휂p2q +2,0 “ ´ +ˆ휂2 +1,0푔p2푘 ´ 1q{푘 +ˆ푆2,0 +, +ˆ휂p2q +0,2 “ +ˆ휂2 +0,1푔푘p푘 ´ 2q +ˆ푆0,2 +, +ˆ휂p2q +1,1 “ ´ ˆ휂1,0 ˆ휂0,1푔p푘 ` 1q +ˆ푆1,1 +, +ˆ휂p2q +1,´1 “ ´ ˆ휂1,0 ˆ휂0,1푔p푘 ` 1q +ˆ푆1,´1 +, +where +(3.33) +ˆ푆푗1,푗2 “ 푔 +푘 p|푗1 ` 푘푗2| ´ 푘qp|푗1 ` 푘푗2| ´ 1q. +Remark 3.3. Even though we stop at the second order in the weakly nonlinear model, one can +continue computing higher-order terms by induction. Suppose that we have obtained terms +of order 푛 ´ 1 for ˜휂 and terms of order 푛 ´ 2 for 푏 and 휏. Eliminating the coefficients of 휖푛 in +(3.11), we find that +(3.34) +푔 ˜휂p푛q ´ 푏p0q퐻coth“ +˜휂p푛q +훼 +‰ +´ 휏p0q ˜휂p푛q +훼훼 ´ 푏p푛´1q퐻coth“ +˜휂p1q +훼 +‰ +´ 휏p푛´1q ˜휂p1q +훼훼 “ 퐶p푛q, +where 퐶p푛q depends on +␣ +푏p푗q( +0ď푗ď푛´2, +␣ +휏p푗q( +0ď푗ď푛´2 and +␣ +˜휂p푗q( +0ď푗ď푛´1. Comparing the Fourier +coefficients of both sides of the above equation, we have +(3.35) +ˆ푆푗1,푗2 ˆ휂p푛q +푗1,푗2 ` ˆ푄p푛´1q +푗1,푗2 +ˆ휂p1q +푗1,푗2 “ ˆ퐶p푛q +푗1,푗2, + +QUASI-PERIODIC WATER WAVES OF FINITE DEPTH +19 +where ˆ푆푗1,푗2 and ˆ푄p푛´1q +푗1,푗2 +are given in (3.28) and (3.29), respectively. Eventually we can express +푏p푛´1q, 휏p푛´1q and the Fourier coefficients of ˜휂p푛q as follows, +(3.36) +ˆ휂p푛q +푗1,푗2 “ +ˆ퐶p푛q +푗1,푗2 +ˆ푆푗1,푗2 +, +p푗1, 푗2q ‰ p˘1, 0q, p0, ˘1q, +푏p푛´1q “ +ˆ퐶p푛q +0,1 +ˆ휂0,1 ´ 푘2 ˆ퐶p푛q +1,0 +ˆ휂1,0 +푘p푘 cothpℎq ´ cothp푘ℎqq, +휏p푛´1q “ +cothpℎq +ˆ퐶p푛q +0,1 +ˆ휂0,1 ´ 푘 cothp푘ℎq +ˆ퐶p푛q +1,0 +ˆ휂1,0 +푘p푘 cothpℎq ´ cothp푘ℎqq +. +Note that the Fourier coefficients of ˜휂p푛q are obtained through a division by ˆ푆푗1,푗2 for p푗1, 푗2q ‰ +p˘1, 0q, p0, ˘1q. If the ˆ푆푗1,푗2 can become arbitrarily small, the corresponding terms ˆ휂p푛q +푗1,푗2 may +be strongly amplified, calling into question the nature of the expansion (3.20). This is known +as a small divisor problem. In the case of deep water, it is clear from (3.33) that some of the +ˆ푆푗1,푗2 approach zero as |푗1|, |푗2| grow without bound. Speculating on the possibilities, it may +be that (3.20) becomes aysmptotic series provided that 푘 is sufficiently irrational, satisfying a +diophantine condition [46] +(3.37) +|푘 ´ 푗1{푗2| ą 퐶|푗2|´휈, +푗1 P Z, 푗2 P Zzt0u, +where 퐶 is a positive constant and 휈 ą 2. But it may also be that exact mathematical solutions +only exist for sufficiently small values of 휖 in a totally disconnected Cantor-like set [37], even +under the assumption (3.37). More research is needed to resolve these questions. +The story is even more complicated in the case where the fluid is of finite depth because +the expression for ˆ푆푗1,푗2 involves the hyperbolic cotangent function. But this formula becomes +simpler again in the case of shallow water, where ℎ is small. Expanding cothpℎq and cothp푘ℎq +in (3.28) in a Laurent expansion about ℎ “ 0, we obtain +(3.38) +ˆ푆푗1,푗2 “ 푔ℎ4 +45 p|푗1 ` 푘푗2|2 ´ 푘2qp|푗1 ` 푘푗2|2 ´ 1q ` 푂pℎ6q. +We notice that ˆ푆푗1,푗2 can be very small due to the factor of ℎ4 in (3.38). Thus, in the shallow water +regime, the amplitudes of quasi-periodic traveling waves bifurcating from the zero-amplitude +solution must be small, with 휖 at most 푂pℎ4q, if weakly nonlinear theory is to predict their +behavior. +4. Numerical methods and results +As in Section 3 above, we focus our discussion on quasi-periodic functions with two quasi- +periods. All computation will be performed with respect to torus functions on T2; the one- +dimensional quasi-periodic functions will be reconstructed from the torus functions using +(2.5). +Let 푓 p훼q be a quasi-periodic function with two quasi-periods and let ˜푓 denote the +corresponding periodic function on T2, +(4.1) +푓 p훼q “ ˜푓 p훼, 푘훼q, +˜푓 p훼1, 훼2q “ +ÿ +푗1,푗2PZ +ˆ푓푗1,푗2푒푖p푗1훼1`푗2훼2q, +p훼1, 훼2q P T2. +Following [73,74], we adopt a pseudo-spectral method and represent ˜푓 in two ways: +(1) Via the values of ˜푓 on a uniform 푀1 ˆ 푀2 grid on the torus T2, +(4.2) +˜푓푚1,푚2 “ ˜푓 p2휋푚1{푀1 , 2휋푚2{푀2q, +0 ď 푚1 ă 푀1 , 0 ď 푚2 ă 푀2. + +20 +J. WILKENING AND X. ZHAO +(2) Via the truncated two-dimensional Fourier series of ˜푓 , with Fourier coefficients given +by +(4.3) +ˆ푓푗1,푗2 “ +1 +푀2 +푀2´1 +ÿ +푚2“0 +˜ +1 +푀1 +푀1´1 +ÿ +푚1“0 +˜푓푚1,푚2푒´2휋푖푗1푚1{푀1 +¸ +푒´2휋푖푗2푚2{푀2, +0 ď 푗1 ď 푀1{2, +´푀2{2 ă 푗2 ď 푀2{2. +We use the ‘r2c’ and ’c2r’ version of the 2d FFTW library to rapidly transform between these +two forms. Products, powers and quotients in (2.54) and (3.11) are evaluated point-wise on the +grid while derivatives and Hilbert transforms are computed in Fourier space via Definition 2.2 +and 2.3. In the scope of this paper, we choose 푘 “ 1{ +? +2 for all numerical examples. +4.1. Time evolution of spatially quasi-periodic waves of finite depth. To compute the time +evolution of spatially quasi-periodic waves, we discretize (2.54) on T2 and use the fifth-order +explicit Runge-Kutta method of Dormand and Prince [32, 74] as the time stepping scheme. +The initial condition of the water wave is given in physical space, which is more natural in +practice, and we compute the conformal mapping to transform the initial condition from +physical space to conformal space using the method described in Appendix A. The numerical +examples discussed below are gravity waves but our numerical method also applies to the +case of nonzero surface tension. +Figure 2. Time evolution of an initially flat free surface in the presence of a background +flow and a quasi-periodic bottom boundary. +Figure 2 shows the time evolution of a free surface wave that is initially flat and develops +quasi-periodic dynamics in the presence of a background flow and a quasi-periodic bottom +boundary. In physical space, the bottom boundary is parameterized by +(4.4) +휂b,physp푥q “ ´1 ` 0.2 cosp푥q ` 0.2 cosp푥{ +? +2q +and the mean velocity of the background flow in (2.29) is U “ 1. In the computation, we +use 푀1 “ 푀2 “ 512 and compute the time evolution of the wave from 푡 “ 0 to 푡 “ 3 with +time steps Δ푡 “ 10´5. In panel (a), the black line plots the bottom boundary and the blue +line plots the flat free surface at 푡 “ 0. To better distinguish the shape of the free surface at + +QUASI-PERIODIC WATER WAVES OF FINITE DEPTH +21 +different times, we plot the free surface at different times with an upward spatial shift. The +time difference between two adjacent curves is 0.06, and we plot +(4.5) +휂sp훼, 푡푛q ` 10푡푛{3, +푡푛 “ 0.06푛, +푛 “ 0, 1, . . . , 50. +Due to the background flow and the quasi-periodic bottom boundary, the free surface wave +moves from left to right and forms wave crests ahead of the peaks of the bottom boundary, +which deflects the fluid upward. Panels (b) and (c) show snapshots of the time evolution of +the free surface from 푡 “ 0 to 푡 “ 1.5 and from 푡 “ 1.5 to 푡 “ 3 separately without the upward +shift given in (4.5); the time difference between two adjacent curves in both panels is 0.15. One +can observe that the free surface gradually develops quasi-periodic crests and troughs, which +drift from left to right due to the background flow; the height difference between crests and +neighboring troughs increases with time. +Figure 3 shows the time evolution of an initially periodic free surface wave in the presence of +a periodic bottom boundary whose spatial period is irrationally related to the initial condition. +In physical space, the initial free surface and the bottom boundary are given by +(4.6) +휂s,phys +0 +p푥q “ 0.2 cosp푥q, +휂b,physp푥q “ ´1 ` 0.2 cosp푥{ +? +2q. +In panel (a), the initial free surface and the bottom boundary are plotted with blue and black +curves, respectively. As shown in the figure, they are both periodic and the bottom boundary’s +wavelength is longer than that of the free surface. We use 푀1 “ 푀2 “ 256 in the computation +and evolve the water wave from 푡 “ 0 to 푡 “ 10 with time steps Δ푡 “ 2 ˆ 10´5. At 푡 “ 0, the +fluid is at rest with zero velocity potential. In panel (a), we plot the time evolution of the free +surface with an upward spatial shift +(4.7) +휂sp훼, 푡푛q ` 0.75푡푛, +푡푛 “ 0.2푛, +푛 “ 0, 1, . . . , 50. +The free surface flattens due to the force of gravity and rises again due to inertia, which is +similar to the oscillation of a standing water wave [44, 69]. One can observe that the crests +and troughs of the surface wave are not symmetric for 푡 ą 0 except at 푥 “ 0 due to the even +symmetry of the initial condition. In panels (b), (c) and (d), we plot snapshots of the time +evolution of the free surface without the upward shit (4.7) from 푡 “ 0 to 푡 “ 3.2; from 푡 “ 3.2 +Figure 3. +Time evolution of an initially periodic free surface in the presence of a +periodic bottom boundary whose spatial period is irrationally related to the period of +the free surface. + +22 +J. WILKENING AND X. ZHAO +Figure 4. +Panel (a) shows the velocity field of the fluid corresponding to the free +surface wave in Figure 2 at 푡 “ 3. +Panel (b) shows the velocity field of the fluid +corresponding to the wave in Figure 3 at 푡 “ 10. The colors represent the magnitude +of the velocity field. +to 푡 “ 7; and from 푡 “ 7 to 푡 “ 10. The time difference between two adjacent curves is 0.2. +One can observe that the wave oscillates up and down like a standing wave. However, as +a consequence of the quasi-periodic interactions between the surface wave and the bottom +boundary, the heights of different crests are different at any given time. +In Figure 4, we plot the velocity field of the waves in Figures 2 and 3 at the final times. The +arrows denote the direction of the velocity field and the colors represent the magnitude of the +velocity field. Panel (a) corresponds to the free surface wave in Figure 2 at 푡 “ 3. One can +observe that at any point in the fluid, the velocity field’s horizontal component is positive: +Φphys +푥 +ą 0, which is consistent with the presence of a background flow from left to right. At the +bottom boundary, the velocity field is parallel to the boundary, which satisfies the Neumann +boundary condition (2.2). Panel (b) shows the velocity field of the fluid corresponding to +the wave in Figure 3 at 푡 “ 10. Unlike panel (a), since there is no background flow, the +direction of the velocity field’s horizontal component varies throughout the fluid, and there +are points where the horizontal component vanishes. The velocity magnitude is relatively +large in the yellow jets, where one can deduce from the direction of the velocity field that +crests are forming. +4.2. Spatially quasi-periodic traveling waves. We formulate the traveling wave problem as a +nonlinear least-squares problem, which we solve using a variant of the Levenberg-Marquardt +algorithm [50, 72, 73]. In Section 3.1, we introduced the residual function R in (3.11), which +depends on 휏, 푏, ˜휂, and demonstrated that the solutions of the traveling wave problem are the +solutions of Rr휏, 푏, ˜휂s “ 0. In the computation, we consider 휏, 푏 and the Fourier coefficients +of ˜휂 as unknowns, denoted ˆ휂, and define the following scalar objective function +(4.8) +Fr휏, 푏, ˆ휂s :“ +1 +8휋2 +ż +T2 R2r휏, 푏, ˆ휂s 푑훼1 푑훼2. + +0 +个个 +-0.5 +个 +-1 +2元 +2 元 +4π +6T +8元 +10元 +012 +10 +8 +6 +4 +20 +→ +-0.5 +个 +→ +个 +2元 +0 +2 元 +4π +6元 +8元 +10元0.6 +0.4 +0.2 +0QUASI-PERIODIC WATER WAVES OF FINITE DEPTH +23 +Note that solving (3.11) is equivalent to finding a zero of the objective function Fr휏, 푏, ˆ휂s. +For the unknown ˆ휂, we only vary the leading Fourier coefficients ˆ휂푗1,푗2 with |푗1| ď 푁1 ă +푀1{2, |푗2| ď 푁2 ă 푀2{2 and set the other Fourier coefficients to zero. +According to the +assumption (3.15), we also set ˆ휂0,0 “ 0 and require that the Fourier coefficients ˆ휂푗1,푗2 are +real and satisfy ˆ휂´푗1,´푗2 “ ˆ휂푗1,푗2 . Consequently, the number of independent leading Fourier +coefficients is +(4.9) +푁tot “ 푁1p2푁2 ` 1q ` 푁2. +As discussed in Section 3.1, we choose ˆ휂1,0, ˆ휂0,1 and ℎ as bifurcation parameters when com- +puting quasi-periodic traveling solutions bifurcating from the zero-amplitude solution and +fix them at nonzero amplitudes in the minimization of F. Therefore there are 푁tot parameters +to compute, which are stored in a vector 풑 as follows +(4.10) +푝1 “ 휏, +푝2 “ ˆ휂1,1, +푝3 “ 푏, +푝4 “ ˆ휂1,´1, +푝5 “ ˆ휂0,2 , . . . , 푝푁tot “ ˆ휂1,´푁2. +The Fourier modes have been organized in a spiral fashion so that low frequency modes +appear first in the list and ˆ휂1,0, ˆ휂0,1 have been replaced by 휏 and 푏; see [73] for details. Our +goal is to find 풑 given ˆ휂1,0 and ˆ휂0,1 such that Fr풑; ˆ휂1,0, ˆ휂0,1s “ 0, where we have re-ordered the +arguments of F and R in (4.8). In the computation, the function R is evaluated at 푀1 ˆ 푀2 +grid points, hence there are 푀1푀2 equations, which are more than the number of unknowns. +For this reason, the nonlinear least-squares problem is overdetermined. +The objective function F is computed from R by the trapezoidal rule approximation over +T2, which is spectrally accurate, +(4.11) +푓 p풑q “ 1 +2푟p풑q푇푟p풑q « F r풑; ˆ휂1,0, ˆ휂0,1s , +푟푚p풑q “ R r풑; ˆ휂1,0, ˆ휂0,1s p훼푚1, 훼푚2q +?푀1푀2 +, +˜ +푚 “ 1 ` 푚1 ` 푀1푚2 +훼푚푖 “ 2휋푚푖{푀푖 +¸ +, +0 ď 푚푖 ă 푀푖. +The parameters 푝푗 are chosen to minimize 푓 p풑q using the Levenberg-Marquardt method [50, +72]. +The method requires a Jacobian matrix B푟푚{B푝푗, which we compute by solving the +variational equations (3.17). We have B푟푚 +B푝푗 “ 훿Rp훼푚1, 훼푚2q{?푀1푀2, where 푚 “ 1`푚1`푀1푚2 +and the 푗th column of the Jacobian corresponds to setting 훿푝푗 in (4.10) to 1 and the others to 0 +depending on the perturbation direction: 훿휏, 훿푏 or 훿 ˆ휂푗1,푗2. +We compute quasi-periodic traveling solutions that bifurcate from the zero solution using +푁푥 “ 푁푦 “ 75 and 푀푥 “ 푀푦 “ 200. We fix ˆ휂1,0 “ ˆ휂0,1 “ 10´5, choose ℎ to be the continuation +parameter, and decrease ℎ from 3 to 0.5 with Δℎ “ 0.01 to obtain a family of quasi-periodic +solutions. In panel (a) of Figure 5, we plot the wave profile of the free surface for solutions +at ℎ “ 0.5 and ℎ “ 3. The difference between these two solutions is small because they are +both small-amplitude bifurcations from the zero solution for which we imposed the same +amplitude parameters ˆ휂1,0 and ˆ휂0,1 at linear order. We stayed close to the linear regime in this +example to investigate whether traveling solutions of the fully nonlinear equations, which we +compute using the Levenberg-Marquardt method, behave as predicted by weakly nonlinear +theory. While the wave profiles are close to one another, the values of ℎ (3 and 0.5) and 휏 +(1.23088845108 and 0.0812490184995) differ substantially for the two solutions. +In panel (b) of Figure 5, we plot the absolute value of the leading Fourier coefficients |ˆ휂2,0|, +|ˆ휂0,2|, |ˆ휂1,1| and |ˆ휂1,´1| of the computed solutions as functions of ℎ, holding ˆ휂1,0 and ˆ휂0,1 fixed +at 10´5. These Fourier coefficients decrease as ℎ increases. In panel (c), we plot the absolute +value of the divisors ˆ푆푗1,푗2 defined in (3.28) corresponding to these four Fourier coefficients, + +24 +J. WILKENING AND X. ZHAO +Figure 5. Quasi-periodic traveling gravity-capillary waves bifurcating from the zero +solution. (a) Surface elevation function of two solutions with ℎ “ 0.5 (dashed red +line) and ℎ “ 3.0 (solid black line). (b) Amplitudes of Fourier coefficients ˆ휂2,0, ˆ휂0,2, +ˆ휂1,1, ˆ휂1,´1 of quasi-periodic traveling solutions for which ˆ휂1,0 and ˆ휂0,1 are fixed at 10´5. +(c) Absolute value of the corresponding divisors ˆ푆푗1,푗2 defined by (3.28) in the weakly +nonlinear model (3.32). Here we solve (3.11) by minimizing 푓 p풑q in (4.11) and check +whether the solution behaves as predicted by (3.32). Panels (d) and (e) show |ˆ휂푗1,푗2| +versus | ˆ푆푗1,푗2| for 2 ď |푗1| ` |푗2| ď 75 in the cases where ℎ “ 0.5 and ℎ “ 3, respectively. +which decrease as ℎ decreases. The behavior of the Fourier coefficients and ˆ푆푗1,푗2 is consistent +with the weakly nonlinear approximations (3.32), where the Fourier coefficients are obtained +through division by ˆ푆푗1,푗2. As a result, smaller values of ˆ푆푗1,푗2 lead to larger Fourier coefficients. +Note that we are checking whether traveling solutions of the Euler equations (3.11) obtained + +10- +10° +20 +10 +-25 +10~6 +10° +100 +102 +10410-10 +10-15 +10-20 +10-25 +10-2 +100 +102 +104QUASI-PERIODIC WATER WAVES OF FINITE DEPTH +25 +by minimizing 푓 p풑q « F r풑; ˆ휂1,0, ˆ휂0,1s in (4.11) via the Levenberg-Marquardt method behave +as predicted by the weakly nonlinear model (3.32); we did not solve (3.32) directly. +Panels (d) and (e) of Figure 5 demonstrate the relationship between |ˆ휂푗1,푗2| and | ˆ푆푗1,푗2| with +2 ď |푗1| ` |푗2| ď 75 for traveling solutions at ℎ “ 0.5 and ℎ “ 3, respectively. Since the largest +Fourier coefficients are fixed at 10´5, one expects roundoff errors around 10´20. But instead +the “roundoff floor,” visible in both panels, appears to grow linearly as | ˆ푆푗1,푗2| decreases. +This suggests that roundoff errors in the Levenberg-Marquardt method are amplified by the +reciprocals of the divisors ˆ푆푗1,푗2 even though this is not a weakly nonlinear calculation. The +“active” modes in which +ˇˇˆ휂푗1,푗2 +ˇˇ extends above the roundoff floor appear to be well-resolved. +The plots look nearly identical if we refine the calculation, keeping 푁푥 “ 푁푦 “ 75 but +increasing 푀푥 and 푀푦 from 200 to 300. In fact, we plotted the data from this finer mesh +in panels (d) and (e). +In panel (e), when ℎ “ 3, there are just a few active modes ˆ휂푗1,푗2, +and they all correspond to low frequency modes with 2 ď |푗1| ` |푗2| ď 5. But in panel (d), +when ℎ “ 0.5, there are many active modes of both small and intermediate frequency, plotted +with red and black markers, respectively. This is consistent with (3.38) and panel (c), where +the small divisors from weakly nonlinear theory decrease as ℎ decreases. The fixed values +ˆ휂1,0 “ ˆ휂0,1 “ 10´5 we selected for this calculation appear to be small enough when ℎ “ 3 that +we could have computed the solution by weakly nonlinear theory, but large enough at ℎ “ 0.5 +that it was necessary to solve the problem by the Levenberg-Marquardt approach. +Next we search for quasi-periodic bifurcations from finite-amplitude periodic traveling +waves of finite depth. We use a new procedure, described in detail for the case of deep water +in [75], to locate bifurcation points. Specifically, we use the signed smallest singular value +휒p푠q of the Jacobian J quap푠q, as in (3.18), as a bifurcation “test function” that changes sign at +bifurcation points. When a zero of 휒p푠q is found, the kernel of the Jacobian J quap푠q of (3.18) +also furnishes a search direction 훿 ˜휂qua for the quasi-periodic branch. We use ˜휂per ` 휖훿 ˜휂qua +with an empirically chosen value of 휖 as the initial guess for the Levenberg-Marquardt solver. +We then use numerical continuation to follow this branch beyond the realm of linearization +about the periodic traveling wave. Instead of using ˆ휂1,0, ˆ휂0,1 and ℎ as continuation parameters, +we use 휏, ℎ and the Fourier mode ˆ휂0,1. For simplicity, we hold 휏 and ℎ fixed and just vary the +Fourier mode to obtain a one-parameter family of quasi-periodic solutions. +Figures 6 and 7 show two quasi-periodic gravity-capillary waves bifurcating from a branch +of periodic traveling waves. The fluid depth in conformal space is ℎ “ 0.1. We set 휏 “ +0.00327672209262 so that the first Fourier mode of the periodic waves resonates with the fifth +Fourier mode, which corresponds to solutions of the Wilton ripple problem [1, 3, 63]. For +the periodic traveling wave, we set 푀 “ 300, 푁 “ 100 and use 푠 “ ˆ휂1 as the continuation +parameter. +The 1D waves are computed on T and embedded in T2 when searching for +bifurcations, so that ˆ휂1 becomes ˆ휂1,0. We computed periodic waves with amplitude 푠 ranging +from 10´5 to 2 ˆ 10´4 with Δ푠 “ 10´5. By tracking the sign of 휒p푠q, we find out that there is a +zero of 휒p푠q when 푠 belongs to intervals r10´5, 2ˆ10´5s, r4ˆ10´5, 5ˆ10´5s, r7ˆ10´5, 8ˆ10´5s, +r1.1ˆ10´4, 1.2ˆ10´4s and r1.7ˆ10´4, 1.8ˆ10´4s. We focus our discussion on the first and last +intervals and locate the zeros of 휒p푠q in these intervals, which are the bifurcation points, using +the numerical algorithm described in [75]. In double precision, the zeros and corresponding +values of 휒 are +(4.12) +푠1 “ 1.83810709940 ˆ 10´5, +휒p푠1q “ ´7.8 ˆ 10´15, +푠2 “ 1.72625902886 ˆ 10´4, +휒p푠2q “ 4.8 ˆ 10´15. + +26 +J. WILKENING AND X. ZHAO +The periodic solutions at 푠1 and 푠2 are plotted with dotted black lines in panel (a) of Figures 6 +and 7, respectively. These periodic solutions demonstrate the nonlinear interaction of Fourier +modes of different wavelengths. +Unlike the crests of sinusoidal waves, we observe small +ripples at the wave peaks of the periodic wave at 푠1. As the amplitude of the periodic solution +increases, this nonlinear feature is more pronounced. For the periodic solution at 푠2, near +푥 “ 2휋푛 for 푛 P Z, there is a flat plateau with wave peaks shifted to the edges of the plateau, +forming an interesting “cat ears” structure. These nonlinear features at the wave crests can be +attributed to the effect of the capillary force. In panel (b) of Figures 6 and 7, we show contour +plots of torus functions of these periodic traveling waves. We observe that the width of the +yellow region is larger for the higher-amplitude periodic wave; in correspondence, this wave +possesses wider wave crests. +We compute secondary quasi-periodic bifurcation branches that intersect the primary pe- +riodic branch at 푠1 and 푠2 and show the corresponding results in Figures 6 and 7, respectively. +In both computations, we set 푀푥 “ 300, 푀푦 “ 150, 푁푥 “ 100, 푁푦 “ 50 and use ˆ휂0,1 as the +continuation parameter. We follow the two quasi-periodic branches until ˆ휂0,1 “ 7 ˆ 10´5 +Figure 6. Quasi-periodic bifurcation from a periodic traveling gravity-capillary wave. +Panel (a) shows the periodic traveling wave where a bifurcation was found and the +largest-amplitude solution we computed on the quasi-periodic bifurcation branch. The +dotted black line corresponds to the periodic wave and the red line corresponds to the +quasi-periodic wave. Panels (b) and (c) show contour plots of the torus functions of the +periodic wave and the quasi-periodic wave, respectively. The 1D quasi-periodic wave +in panel (a) is extracted from the corresponding torus function along the characteristic +lines of slope 푘 “ 1{ +? +2, plotted with red dashed lines in panel (c). + +T +2 +0 +0 +-2 +-4 +-T +0 +-T +TTT +2 +0 +0 +-2 +-4 +-T +0 +-T +T +X10'QUASI-PERIODIC WATER WAVES OF FINITE DEPTH +27 +Figure 7. +Quasi-periodic bifurcation from a larger-amplitude periodic traveling +gravity-capillary wave. The panels show the same information as in Figure 6. +and ˆ휂0,1 “ 1.1 ˆ 10´4, respectively; the corresponding quasi-periodic traveling waves are +plotted with red lines in panel (a) of Figures 6 and 7. The objective function is minimized to +2.14 ˆ 10´27 and 5.03 ˆ 10´28, respectively, for these solutions. In panel (a) of Figure 6, the os- +cillations at the troughs of the quasi-periodic wave are ahead of the ones of the periodic wave +near 휉 “ 3휋, 5휋, 7휋, 11휋 and are behind near 휉 “ 휋, which demonstrates the quasi-periodic +feature of the secondary bifurcation. +We also observe that the amplitude of the quasi-periodic wave in panel (a) of Figure 6 +is noticeably larger than the periodic wave due to the activation of Fourier modes in the +quasi-periodic direction. In panel (c) of Figures 6 and 7, we show contour plots of the torus +functions of the quasi-periodic traveling waves in panel (a). Unlike the periodic solution, +the quasi-periodic solution depends on 훼2. For example, one can see the variation of the +yellow and blue regions in the 훼2 direction. Moreover, this variation is rather oscillatory in +Figure 6, which adds to the difficulty of computing higher-amplitude quasi-periodic waves +on the bifurcation branch. +The 1D quasi-periodic waves are obtained by evaluating the +corresponding torus functions along the the red dashed line of slope 1{ +? +2. In panel (a) of +Figures 6 and 7, there will be crests if the dashed line in panel (c) passes through the yellow +region and troughs if it passes through the blue region. Due to the variation in yellow region, +the widths of the crests of the quasi-periodic wave are no longer constant. For example, in +panel (a) of Figure 7, the crests of the quasi-periodic wave are wider than those of the periodic +wave near 휉 “ 6휋, 8휋 and narrower near 휉 “ 4휋, 10휋. + +T +2 +0 +-2 +0 +-4 +9- +-8 +-10 +-T +0 +-T +T +X10T +2 +0 +-2 +0 +-4 +-6 +-T +0 +-T +T +X1028 +J. WILKENING AND X. ZHAO +5. Conclusion +In this paper, we have presented a numerical study of two-dimensional finite-depth free +surface waves in the spatially quasi-periodic setting. Specifically, we have studied both the +initial value and traveling wave problems. +For the initial value problem, we derived the +governing equations of water waves in the presence of a background flow and a non-flat +bottom boundary in conformal space. As noted in Remark 2.7, the derivation is valid in both +the quasi-periodic and periodic settings. Motivated by the experiments of Torres et al. [62] +studying spatially quasi-periodic surface waves in the presence of a quasi-periodic bottom +boundary, we computed the time evolution of an initially flat surface with a background flow +over a quasi-periodic bottom boundary. We also find that the waves develop quasi-periodic +patterns in which the distance between adjacent wave peaks is not constant. +Next we computed spatially quasi-periodic traveling waves that bifurcate from the zero- +amplitude wave or from finite-amplitude periodic traveling waves. Motivated by observations +in [73, 75] that the Fourier coefficients of quasi-periodic traveling waves decay slower along +certain directions, we derived the weakly nonlinear equations governing small-amplitude +quasi-periodic traveling waves in Section 3.2 and found that there is a divisor ˆ푆푗1,푗2 in the +formula for the Fourier coefficients ˆ휂푗1,푗2 of the weakly nonlinear solutions. For example, in +the case of deep water, this divisor reads +(5.1) +ˆ푆푗1,푗2 “ 푔 +푘 p|푗1 ` 푘푗2| ´ 푘qp|푗1 ` 푘푗2| ´ 1q. +Due to the unboundedness of 1{ ˆ푆푗1,푗2, the Fourier coefficients along directions |푗1 ` 푘푗2|´ 푘 “ 0 +and |푗1 ` 푘푗2| ´ 1 “ 0 are expected to decay slower than in other directions. We also study +these divisors in the case of shallow water and find that weakly nonlinear theory breaks down +faster when ℎ is smaller due to the factor of ℎ4 in the formula (3.38) for ˆ푆푗1,푗2. +In the current work, we assume that the bottom boundary remains fixed in time. In the +future, we plan to further extend our method to study quasi-periodic flows with a free surface +over a moving bottom boundary. In the case of periodic water waves, this has been studied +in [57, 59]. We also plan to analyze the linear stability of periodic traveling waves [19, 47, 52, +60] and investigate the long-time dynamics of traveling waves under unstable subharmonic +perturbations. In the quasi-periodic setting, we are able to compute the exact time evolution of +these perturbed waves instead of their linearized approximations [38]. We are also interested in +developing numerical methods, such as the Transformed Field Expansion method [48,49,54], +to study the dynamics of these waves in three dimensions where the conformal mapping +method no longer applies. On the theoretical side, a rigorous proof of the existence of quasi- +periodic traveling waves is still an open problem. We expect it will be necessary to employ a +Nash-Moser iteration to tackle the small divisor problem, which has been successfully used to +prove the existence of temporally quasi-periodic standing waves and traveling waves [9,10]. +Funding: This work was supported in part by the National Science Foundation under award +number DMS-1716560 and by the Department of Energy, Office of Science, Applied Scientific +Computing Research, under award number DE-AC02-05CH11231; and by an NSERC (Canada) +Discovery Grant. +Declaration of interests: The authors report no conflict of interest. + +QUASI-PERIODIC WATER WAVES OF FINITE DEPTH +29 +Appendix A. Computation of the conformal mapping from a infinite horizontal strip to +the fluid domain +In practice, the initial condition of the water wave is usually given in physical space. There- +fore, we need to compute the conformal mapping 푧p푤, 푡q to transform the initial condition +from physical space to conformal space. As shown in (2.23) and (2.24), the conformal mapping +is determined by ℎ, 푥0, ˜휂s and ˜휂b, where 푥0 is fixed to be zero in the scope of this paper and ℎ, +˜휂s, ˜휂b are obtained by solving the following equations, +(A.1) +R1p훼1, 훼2q “ ˜휂s ´ ˜휂s,physp훼1 ` ˜휉s, 훼2 ` 푘 ˜휉sq “ 0, +R2p훼1, 훼2q “ ˜휂b ´ ˜휂b,physp훼1 ` ˜휉b, 훼2 ` 푘 ˜휉bq “ 0, +˜휉s “ 퐻cothr˜휂ss ` 퐻cschr˜휂bs, +˜휉b “ ´퐻cschr˜휂ss ´ 퐻cothr˜휂bs, +which come from (2.17) and (2.26). Moreover, we enforce the constraint ℎ “ ˆ휂s +0 ´ ˆ휂b +0 discussed +in Section 2.3 and rewrite ˜휂b as +(A.2) +˜휂b “ ˆ휂s +0 ´ ℎ ` 푃r˜휂bs. +Otherwise problem (A.1) is underdetermined and the solution is not unique. +In our computations, we consider ℎ and the Fourier coefficients of ˜휂s and ˜휂b as unknowns +and define the following objective function +(A.3) +Frℎ, ˆ휂s, ˆ휂bs : “ +1 +8휋2 +ż +T2 R2 +1rℎ, ˆ휂s, ˆ휂bs ` R2 +2rℎ, ˆ휂s, ˆ휂bs 푑훼1 푑훼2 +« +1 +2푀1푀2 +푀2´1 +ÿ +푚2“0 +푀1´1 +ÿ +푚1“0 +” +R2 +1p2휋푚1{푀1, 2휋푚2{푀2q ` R2 +1p2휋푚1{푀1, 2휋푚2{푀2q +ı +. +We apply a Leveberg-Marquardt method [72] to solve the nonlinear least-squares prob- +lem (A.3) and compute the derivative of R1 and R2 with respect to the unknowns using +the following variational equations +(A.4) +훿R1 “ 훿 ˜휂s ´ ˜휂s,phys +푥 +훿 ˜휉s, +훿R2 “ 훿 ˆ휂s +0 ´ 훿ℎ ` 푃r훿 ˜휂bs ´ ˜휂b,phys +푥 +훿 ˜휉b, +훿 ˜휉s “ 퐻cothr훿 ˜휂ss ` 퐻cschr훿 ˜휂bs ` +` +훿퐻coth˘ +r˜휂ss ` +` +훿퐻csch˘ +r˜휂bs, +훿 ˜휉b “ ´퐻cschr훿 ˜휂ss ´ 퐻cothr훿 ˜휂bs ´ +` +훿퐻csch˘ +r˜휂ss ´ +` +훿퐻coth˘ +r˜휂bs. +Here B푥 “ B푥1 ` 푘B푥2 and the symbols of 훿퐻coth and 훿퐻csch are +(A.5) +훿 ˆ퐻coth +푗1,푗2 “ +푖p푗1 ` 푘푗2q훿ℎ +sinh2pp푗1 ` 푘푗2qℎq +, +훿 ˆ퐻csch +푗1,푗2 “ ´푖p푗1 ` 푘푗2q cothpp푗1 ` 푘푗2qℎq cschpp푗1 ` 푘푗2qℎq훿ℎ. +References +[1] B. 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J. +Fluid Mech., 184:183–206, 1987. + diff --git a/LNAzT4oBgHgl3EQfVvxj/content/tmp_files/load_file.txt b/LNAzT4oBgHgl3EQfVvxj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..85b4dc715cfb8a8b5843f5b3969d85e08cd2912c --- /dev/null +++ b/LNAzT4oBgHgl3EQfVvxj/content/tmp_files/load_file.txt @@ -0,0 +1,1597 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf,len=1596 +page_content='SPATIALLY QUASI-PERIODIC WATER WAVES OF FINITE DEPTH JON WILKENING AND XINYU ZHAO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We present a numerical study of spatially quasi-periodic water waves of finite depth in both the initial value problem and traveling wave settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We adopt a quasi-periodic conformal mapping formulation of the Euler equations, where one-dimensional quasi-periodic functions are represented by periodic functions on a higher-dimensional torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We compute the time evolution of free surface waves in the presence of a background flow and a quasi-periodic bottom boundary and observe the formation of quasi-periodic patterns on the free surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Two types of quasi-periodic traveling waves are computed: small-amplitude waves bifurcating from the zero-amplitude solution and larger-amplitude waves bifurcating from finite-amplitude periodic traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We derive weakly nonlinear approximations of the first type and investigate the associated small-divisor problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We find that waves of the second type exhibit striking nonlinear behavior, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=', the peaks and troughs are shifted non-periodically from the corresponding periodic waves due to the activation of quasi-periodic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Introduction Free surface waves on incompressible fluids arise in many contexts in fluid dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Ex- amples include ocean wave forecasting [38, 61], modeling the motion of flows over obstacles and varying bottom boundaries [5, 29, 67], and studying wind-wave interactions in extreme wave events, such as freak waves [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' These models are described by the Euler equations, which are usually studied under periodic boundary conditions or the assumption that solu- tions decay to zero at infinity [4,33,39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' However, these assumptions are insufficient in many problems of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For instance, a periodic wave could interact with a bottom boundary with a different spatial period, or subharmonic perturbations of a periodic traveling wave can grow in amplitude, leading to quasi-periodic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' To tackle these issues, we recently proposed methods [73, 74] to study the Euler equations under quasi-periodic boundary conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' specifically, we studied spatially quasi-periodic waves of infinite depth in two dimensions and developed numerical algorithms to compute such waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In this paper, we extend this previous work to the finite-depth case and discuss both the initial value and traveling wave problems in the quasi-periodic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Finite-depth water waves exhibit interesting nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' It has been shown nu- merically that Fermi-Pasta-Ulam recurrence can occur in free surface waves of finite depth when the wave amplitude is less than about 1{10 of the fluid depth [56,58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' A varying bottom boundary can lead to substantial amplifications of water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' There have been both ex- perimental and numerical studies demonstrating increased freak wave activities when waves propagate over a sloping bottom, from a deeper to a shallower domain [21,64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the problem of long waves approaching vertical walls, an abrupt transition in the bottom boundary can cause large runups on the wall or wave breaking, which generally occurs when the wave crest Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, USA Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1 E-mail address: wilkening@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='edu, zhaox171@mcmaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='01289v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='flu-dyn] 3 Jan 2023 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' WILKENING AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ZHAO overturns [34, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The interaction between a rotational wave current and a varying bottom boundary gives rise to a time-dependent Kelvin cat-eye structure [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The quasi-periodic dynamics of water waves have recently drawn considerable attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Damanik and Goldstein [18] proved the global existence and uniqueness of small-amplitude spatially quasi-periodic solutions of the KdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Oh [51] and Dodson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' [20] showed the local existence of spatially quasi-periodic solutions of nonlinear Schrödinger equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Berti and Montalto [10] and Baldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' [7] used Nash-Moser theory to prove the existence of small-amplitude temporally quasi-periodic gravity-capillary and gravity standing waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Berti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' [8, 9] and Feola and Giuliani [27] have proved the existence of temporally quasi- periodic gravity-capillary and gravity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' On the numerical side, Wilkening computed new families of relative-periodic [70] and traveling-standing [71] water wave solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We were originally motivated by the structure of quasi-crystals in material science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Bli- nov [11] used quasi-periodic solutions of the Schrödinger equation to describe the electronic structure of non-interacting electrons of quasi-crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' To study how electrons move through quasi-crystals, Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' [62] created quasi-periodic standing waves by vibrating a fluid- filled pan with a quasi-periodic bottom boundary and sent a transverse wave pulse across the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' They observed that the traveling wave pulse demonstrated a non-periodic pattern: the spacing between the wave peaks was not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Their observation inspired us to ask the following question: how do we compute the exact dynamics of free surface waves in the presence of a quasi-periodic bottom boundary?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' To address this question, one needs to study the free surface wave problem in a quasi-periodic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Another reason for our interest in quasi-periodic water waves originates from the dispersion relation of gravity-capillary waves of finite depth: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1) 푐2 “ p푔푘´1 ` 휏푘q tanhp푘ℎq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Here 푐 is the phase speed, 푘 is the wave number, 푔 is the acceleration due to gravity, 휏 is the coefficient of surface tension and ℎ is the depth of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' It is known [63] that when 휏{p푔ℎ2q ă 1{3, there exists 푐crit between 0 and a 푔ℎ such that for any fixed phase speed 푐 ą 푐crit, there are two distinct positive wave numbers satisfying the dispersion relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1), which we denote by 푘1 and 푘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Any superposition of waves with these two wave numbers is a solution of the linearized traveling wave problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' If 푘1 and 푘2 are rationally related, the linear solution is spatially periodic and related to the well-studied Wilton ripples [1,2,63,76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' On the other hand, if 푘1 and 푘2 are irrationally related, the linear solution will be spatially quasi-periodic, which gives a natural place to search for nonlinear quasi-periodic traveling solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Bridges and Dias [14] first studied these spatially quasi-periodic traveling waves using a spatial Hamiltonian structure and constructed weakly nonlinear approximations of these waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Later we [73] used a conformal mapping formulation of the water wave equations and computed highly accurate numerical solutions of the fully nonlinear problem in the case of deep water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the present work, we aim to further extend these techniques to the case of finite-depth water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Following [73, 74], we adopt a conformal mapping formulation of the free surface Euler equations [16,22–25,36,42,78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the finite depth case, the fluid domain with a curved surface and an uneven bottom boundary is mapped conformally onto a horizontal strip instead of the lower half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Since the conformal mapping depends on time, even though the bottom boundary is fixed in physical space, the representation of the bottom boundary in conformal space varies with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Ruban [55, 57] fixed the width of the strip and used a composition of two conformal mappings to map the strip to the fluid domain – the first leaves the real QUASI-PERIODIC WATER WAVES OF FINITE DEPTH 3 axis invariant and the second maps the real line to the bottom boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Viotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' [67] and Flamarion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' [30] let the width of the strip vary with time to keep the wave length the same in physical space and conformal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' They used a fixed-point iterative method to compute the bottom profile at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In order that water waves possess the same quasi-periods in both physical and conformal spaces, we also let the strip width be a time-dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' However, in contrast to [30, 67], we compute the time evolution of the bottom profile directly, employing analytical properties of the conformal mapping, similar to [55, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' As in the infinite-depth case [73, 74], we introduce quasi-periodic Hilbert transforms to relate the real and imaginary parts of the conformal mapping and to compute the kinematic boundary condition on the free surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' These Hilbert transforms are Fourier multiplier operators and are easier to compute in a quasi-periodic setting than a more direct computation of the Dirichlet-Neumann operator [17] in physical space, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=', using boundary integral methods [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In computing the dynamics of free surface waves over a varying bottom boundary, it is usually assumed that the spatial periods of the free surface wave and the bottom boundary are the same or one is an integer multiple of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In this paper, we study a new situation where their spatial periods are irrationally related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Specifically, in one of the examples presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1, we compute the time evolution of an initially periodic free surface wave with period 2휋 in the presence of a periodic bottom boundary with period 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 2휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We find that the periodic wave becomes a quasi-periodic wave, with each wave peak and trough evolving differently as it interacts with the bottom boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' By the time-reversibility of the Euler equations, we learn that a quasi-periodic wave can evolve to a periodic wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We also compute the time evolution of an initially flat free surface in the presence of a background flow and a quasi-periodic bottom boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Similar to the experiment by Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' [62], we also observe that the free surface wave develops quasi-periodic patterns as a result of interactions between the background flow and the quasi-periodic bottom boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The wave peaks and troughs are asymmetric and the distance between adjacent wave peaks is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2, we compute two types of quasi-periodic traveling solutions: waves that bifurcate from the zero-amplitude solution and waves that bifurcate from finite-amplitude periodic traveling solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For the first type, we use linearization about the zero solution for the initial bifurcation direction and obtain a three-parameter family of solutions prescribed by the fluid depth and Fourier coefficients corresponding to wave numbers 푘1 and 푘2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' these are called the base Fourier coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Similar to the case of deep water [73], when the amplitudes of the base Fourier coefficients are small, the solutions are of small amplitude and are close to the linear solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For the second type, we linearize the governing equations around a finite-amplitude 2휋-periodic traveling wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For the bifurcation direction in this case, we use a quasi-periodic function of the following form in the kernel of the linear operator: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2) 훿휂p훼q “ 푒푖푘훼휂0p훼q ` 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=', where 휂0 possesses the same wavelength as the periodic traveling wave, the notation 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' denotes the complex conjugate of the preceding term, and we set 푘 “ 1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 2 in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' This method has also been used to compute secondary periodic bifurcations with 푘 “ 1{2 and 푘 “ 1{3 by Chen and Saffman [15] and with 푘 “ 1{9 by Vanden-Broeck [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the present work, we obtain quasi-periodic traveling waves that bifurcate from a periodic traveling wave whose first Fourier mode resonates with the fifth Fourier mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The periodic traveling wave is a solution of the Wilton ripple problem and the wave peaks look like “cat ears”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The bifurcated 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' WILKENING AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ZHAO wave still preserves this characteristic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' however, influenced by the Fourier modes in the quasi- periodic direction, the distance between the successive “ears” is no longer constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In Section 2, we define finite-depth quasi-periodic Hilbert transforms and derive equations of motion for quasi-periodic free surface waves in conformal space when the bottom boundary is not necessarily flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In Section 3, we obtain the governing equations of quasi-periodic traveling waves in the case of finite-depth water with a flat bottom boundary and establish weakly nonlinear approximations of these waves and the role of small divisors in computing successive approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In Section 4, we use a Fourier pseudo-spectral method to compute solutions of the initial value and traveling wave problems and present various numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Following the idea in [73,74], we lift the one-dimensional quasi-periodic problem to a higher-dimensional periodic torus where the computation is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We formulate the traveling wave problem as an overdetermined nonlinear least-squares problem that we solve through a variant of the Levenberg-Marquardt method [50, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For the initial value problem, we consider the natural setting where the quasi-periodic initial condition and bottom boundary are posed in physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We present a method of transforming them to conformal space in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Equations of motion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Governing equations in physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Gravity-capillary waves of finite depth are gov- erned by the free-surface Euler equations [39,77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In two dimensions, they may be written (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1) 휂sp푥, 0q “ 휂s 0p푥q, 휑p푥, 0q “ 휑0p푥q, 푡 “ 0, 푥 P R, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2) Φ푥푥 ` Φ푦푦 “ 0, 휂bp푥q ă 푦 ă 휂sp푥, 푡q, Φ “ 휑, 푦 “ 휂sp푥, 푡q, ∇Φ ¨ 풏 “ 0, 푦 “ 휂bp푥q, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3) 휂s 푡 “ Φ푦 ´ 휂s 푥Φ푥, 푦 “ 휂sp푥, 푡q, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='4) 휑푡 “ Φ푦휂s 푡 ´ 1 2Φ2 푥 ´ 1 2Φ2 푦 ´ 푔휂s ` 휏 휂s 푥푥 ` 1 ` p휂s 푥q2˘3{2 ` 퐶p푡q, 푦 “ 휂sp푥, 푡q, where 푥 is the horizontal coordinate, 푦 is the vertical coordinate, 푡 is the time, Φp푥, 푦, 푡q is the velocity potential of the fluid, 휂sp푥, 푡q is the free surface elevation, 휂bp푥q is the fixed bottom profile, 푔 is the vertical acceleration due to gravity, and 휏 is the coefficient of surface tension, which is zero for gravity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3) is the kinematic boundary condition and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='4) is the dynamic boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The function 퐶p푡q in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='4) is an arbitrary integration constant that is allowed to depend on time but not space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We are interested in the dynamics of the water waves in the presence of a varying bottom boundary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' in other words, the bottom profile is not a constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' When the bottom boundary is flat, it is usually assumed that there is no background flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Indeed, in this case, the system is Galilean invariant, which means any background flow can be eliminated by viewing the system in a moving frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' However, this is not true when the bottom boundary is variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' the interaction between the background flow and the bottom boundary can lead to interesting nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Therefore, it is meaningful to incorporate a background flow in the problem description by including a secular growth term in the velocity potential, which is otherwise spatially periodic or quasi-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' QUASI-PERIODIC WATER WAVES OF FINITE DEPTH 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Quasi-periodic Hilbert transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' As defined in [26,46], a quasi-periodic, real-analytic function 푓 p훼q is a function of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5) 푓 p훼q “ ˜푓 p풌훼q, ˜푓 p휶q “ ÿ 풋PZ푑 ˆ푓풋푒푖x풋, 휶y, 훼 P R, 휶, 풌 P R푑, where x¨, ¨y denotes the standard inner product on R푑 and ˜푓 is a periodic, real-analytic function defined on the 푑-dimensional torus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='6) T푑 :“ R푑L p2휋Zq푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We assume that 푑 ě 2 so that 푓 can be genuinely quasi-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Entries of the vector 풌 are called the basic wave numbers (or basic frequencies) of 푓 and are required to be linearly independent over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Given a quasi-periodic function 푓 , the corresponding ˜푓 and 풌 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5) are not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Indeed, if 푲 is any 푑-by-푑 unimodular matrix, then ˜푓 1p휶q “ ˜푓 p푲휶q also satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5) with 풌1 “ 푲´1풌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For simplicity, we assume 풌 is given, along with 푓 or ˜푓 , to pin down the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Given 풌, one can reconstruct ˜푓 and its Fourier coefficients ˆ푓풋 from 푓 via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='7) ˆ푓풋 “ lim 푎Ñ8 1 2푎 ż 푎 ´푎 푓 p훼q푒´푖x풋,풌y훼푑훼, 풋 P Z푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We refer to [6,12,28,35,46] for detailed discussions of the above averaging formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We assume that ˜푓 p휶q is real-analytic, which is equivalent to the conditions that ˆ푓´풋 “ ˆ푓풋 for 풋 P Z푑 and there exist positive numbers 푀 and 훾 such that | ˆ푓풋| ď 푀푒´훾}풋}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=', the Fourier modes ˆ푓풋 decay exponentially as }풋} Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Next we introduce some operators that act on 푓 and ˜푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The projection operators 푃 and 푃0 are defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='8) 푃 “ id ´푃0, 푃0r 푓 s “ 푃0r ˜푓 s “ ˆ푓0 “ 1 p2휋q푑 ż T푑 ˜푓 p휶q 푑훼1 ¨ ¨ ¨ 푑훼푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Note that 푃 projects onto the space of zero-mean functions and 푃0 returns the mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' There are two versions of 푃 and 푃0, one acting on quasi-periodic functions defined on R and one acting on torus functions defined on T푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The derivative operator B훼 that acts on 푓 or ˜푓 is defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='9) B훼 푓 p훼q “ B훼 ˜푓 p풌훼q, B훼 ˜푓 p휶q “ ÿ 풋‰0 푖x풋, 풌y ˆ푓풋푒푖x풋,휶y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For simplicity of notation, we denote B훼 푓 (or B훼 ˜푓 ) by 푓훼 (or ˜푓훼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' One can also interpret B훼 ˜푓 as the directional derivative of ˜푓 along the characteristic direction 풌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We introduce four quasi-periodic Hilbert transforms 퐻tanh, 퐻coth, 퐻csch, 퐻sech that act on 푓 and ˜푓 as follows (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='10) 퐻tanhr 푓 sp훼q “ 퐻tanhr ˜푓 sp풌훼q, 퐻tanhr ˜푓 sp휶q “ ÿ 풋‰0 푖 tanh ` x풋, 풌yℎ ˘ ˆ푓풋푒푖x풋,휶y, 퐻cothr 푓 sp훼q “ 퐻cothr ˜푓 sp풌훼q, 퐻cothr ˜푓 sp휶q “ ÿ 풋‰0 p´푖q coth ` x풋, 풌yℎ ˘ ˆ푓풋푒푖x풋,휶y, 퐻sechr 푓 sp훼q “ 퐻sechr ˜푓 sp풌훼q, 퐻sechr ˜푓 sp휶q “ ÿ 풋‰0 sech ` x풋, 풌yℎ ˘ ˆ푓풋푒푖x풋,휶y, 퐻cschr 푓 sp훼q “ 퐻cschr ˜푓 sp풌훼q, 퐻cschr ˜푓 sp휶q “ ÿ 풋‰0 p푖q csch ` x풋, 풌yℎ ˘ ˆ푓풋푒푖x풋,휶y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' WILKENING AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ZHAO Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The time-dependent conformal mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Here ℎ is a positive parameter that will be discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The symbols of these Hilbert transforms are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11) ˆ퐻tanh 풋 “ 푖 tanh ` x풋, 풌yℎ ˘ , ˆ퐻coth 풋 “ # p´푖q coth ` x풋, 풌yℎ ˘ , 풋 ‰ 0, 0 풋 “ 0, ˆ퐻sech 풋 “ sech ` x풋, 풌yℎ ˘ , ˆ퐻csch 풋 “ # 푖 csch ` x풋, 풌yℎ ˘ 풋 ‰ 0, 0 풋 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We notice that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='12) lim ℎÑ8 ˆ퐻tanh 풋 “ 푖 sgn ` x풋, 풌y ˘ , lim ℎÑ8 ˆ퐻coth 풋 “ ´푖 sgn ` x풋, 풌y ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The latter coincides with the quasi-periodic Hilbert transform introduced in [73,74] in the case of deep water while the former is its pseudo-inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The quasi-periodic conformal mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Figure 1 illustrates a time-dependent conformal mapping (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='13) 푧p푤, 푡q “ 푥p훼, 훽, 푡q ` 푖푦p훼, 훽, 푡q, 푤 “ 훼 ` 푖훽, that maps the infinite strip in the complex plane (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='14) 푆ℎ “ t훼 ` 푖훽 : 훼 P R, ´ℎp푡q ă 훽 ă 0u to the fluid domain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='15) Ω 푓 “ tp푥, 푦q : 푥 P R, 휂b,physp푥q ă 푦 ă 휂s,physp푥, 푡qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' To avoid ambiguity, we use 휂s,phys and 휂b,phys to denote the free surface elevation and the bot- tom profile in physical space, respectively, whereas 휂s and 휂b are used as conformal variables henceforth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We assume that 푧p푤, 푡q can be extended continuously to 푆ℎ and maps the top and bottom boundary of the strip to the free surface and the bottom boundary of the fluid domain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Denoting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='16) 휁sp훼, 푡q “ 푧|훽“0p훼, 푡q “ 푥p훼, 0, 푡q ` 푖푦p훼, 0, 푡q “ 휉sp훼, 푡q ` 푖휂sp훼, 푡q, 휁bp훼, 푡q “ 푧|훽“´ℎp푡qp훼, 푡q “ 푥p훼, ´ℎp푡q, 푡q ` 푖푦p훼, ´ℎp푡q, 푡q “ 휉bp훼, 푡q ` 푖휂bp훼, 푡q, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='17) 휂s,physp휉sp훼, 푡q, 푡q “ 휂sp훼, 푡q, 휂b,physp휉bp훼, 푡qq “ 휂bp훼, 푡q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' QUASI-PERIODIC WATER WAVES OF FINITE DEPTH 7 For later use in the derivation of the governing equations in conformal space, we compute the derivative with respect to 훼 and 푡 on both sides of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='16) and obtain that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='18) 푥훼 “ 휉s 훼, 푦훼 “ 휂s 훼, 푥푡 “ 휉s 푡 , 푦푡 “ 휂s 푡, p훽 “ 0q as well as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='19) 푥훼 “ 휉b 훼, 푦훼 “ 휂b 훼, 푦훼ℎ푡 ` 푥푡 “ 휉b 푡 , ´푥훼ℎ푡 ` 푦푡 “ 휂b 푡 , p훽 “ ´ℎp푡qq where we use the Cauchy-Riemann relation 푥훼 “ 푦훽 and 푦훼 “ ´푥훽 in the last two equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The derivative of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='17) with respect to 훼 and 푡 yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='20) 휂s,phys 푥 휉s 훼 “ 휂s 훼, 휂s,phys 푥 휉s 푡 ` 휂s,phys 푡 “ 휂s 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='21) 휂b,phys 푥 휉b 훼 “ 휂b 훼, 휂b,phys 푥 휉b 푡 “ 휂b 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We are interested in the case where 휂s and 휂b are quasi-periodic functions of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='22) 휂sp훼, 푡q “ ˜휂sp풌훼, 푡q, ˜휂sp휶, 푡q “ ÿ 풋PZ푑 ˆ휂s 풋p푡q푒푖x풋,휶y, 휂bp훼, 푡q “ ˜휂bp풌훼, 푡q, ˜휂bp휶, 푡q “ ÿ 풋PZ푑 ˆ휂b 풋 p푡q푒푖x풋,휶y, 훼 P R, 휶, 풌 P R푑, where 풌 is fixed and its components are linearly independent over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' This is different from the usual conformal mapping framework [22,23,42,43,45,78], where 휂s and, if present, 휂b are assumed to be periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Using the fact that 푦 is a harmonic function defined on 푆ℎ and the boundary values of 푦 are given by 푦|훽“0 “ 휂s and 푦|훽“´ℎ “ 휂b, we obtain that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='23) 푦 “ 1 ℎ ` ˆ휂s 0 ´ ˆ휂b 0 ˘ 훽 ` ˆ휂s 0 ` ÿ 풋‰0 sinh ` x풋, 풌yp훽 ` ℎq ˘ sinh ` x풋, 풌yℎ ˘ ˆ휂s 풋푒푖x풋,풌y훼 ´ ÿ 풋‰0 sinh ` x풋, 풌y훽 ˘ sinh ` x풋, 풌yℎ ˘ ˆ휂b 풋 푒푖x풋,풌y훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The harmonic conjugate of 푦, which is 푥, can be computed from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='23) using the Cauchy- Riemann equations 푥훼 “ 푦훽, 푥훽 “ ´푦훼, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='24) 푥 “ 1 ℎ ` ˆ휂s 0´ ˆ휂b 0 ˘ 훼`푥0´ ÿ 풋‰0 푖 cosh ` x풋, 풌yp훽 ` ℎq ˘ sinh ` x풋, 풌yℎ ˘ ˆ휂s 풋푒푖x풋,풌y훼 ` ÿ 풋‰0 푖 cosh ` x풋, 풌y훽 ˘ sinh ` x풋, 풌yℎ ˘ ˆ휂b 풋 푒푖x풋,풌y훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Here 푥0 is an integration constant, depending on time only, that we are free to choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Given Ω 푓 at any time, to fix the mapping 푧, we need to specify two free parameters: ℎ and 푥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We set (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='25) ℎ “ ˆ휂s 0 ´ ˆ휂b 0, 푥0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Hence, the first terms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='23) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='24) are just 훼 and 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' One can choose ℎ in the same way when the fluid domain is periodic in 푥 so that wavelengths do not change under the conformal mapping [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Alternatively, one may set ℎ “ 1, as is done in [55, 57] in the periodic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Setting 푥0 “ 0 requires a certain choice to be made for a parameter in the time evolution equations [74];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' this will be discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Until then, we leave 푥0p푡q in the representation as a time-dependent parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' WILKENING AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ZHAO Comparing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='23) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='24), we notice that the values of 푥 and 푦 at the top and bottom boundary of 푆ℎ are related by the quasi-periodic Hilbert transforms of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='10), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='26) 휉sp훼, 푡q “ 훼 ` 푥0p푡q ` 퐻cothr휂ssp훼, 푡q ` 퐻cschr휂bsp훼, 푡q, 휉bp훼, 푡q “ 훼 ` 푥0p푡q ´ 퐻cschr휂ssp훼, 푡q ´ 퐻cothr휂bsp훼, 푡q, 휂s 훼p훼, 푡q “ 퐻cothr휉s 훼sp훼, 푡q ` 퐻cschr휉b 훼sp훼, 푡q, 휂b 훼p훼, 푡q “ ´퐻cschr휉s 훼sp훼, 푡q ´ 퐻cothr휉b 훼sp훼, 푡q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The quasi-periodic complex velocity potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Let Φphysp푥, 푦, 푡q denote the velocity potential in physical space from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1 and let 푊physp푥 ` 푖푦, 푡q “ Φphysp푥, 푦, 푡q ` 푖Ψphysp푥, 푦, 푡q be the complex velocity potential, where Ψphys is the stream function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Us- ing the conformal mapping (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='13), we pull back these functions to the strip 푆ℎ and define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='27) 푊p푤, 푡q “ Φp훼, 훽, 푡q ` 푖Ψp훼, 훽, 푡q “ 푊physp푧p푤, 푡q, 푡q, 푤 “ 훼 ` 푖훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We denote 휑s “ Φ|훽“0, 휑b “ Φ|훽“´ℎ, 휓s “ Ψ|훽“0, 휓b “ Ψ|훽“´ℎ and use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='16) to obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='28) 휑sp훼, 푡q “ Φphysp휉sp훼, 푡q, 휂sp훼, 푡q, 푡q “ 휑s,physp휉sp훼, 푡q, 푡q, 휑bp훼, 푡q “ Φphysp휉bp훼, 푡q, 휂bp훼, 푡q, 푡q, 휓sp훼, 푡q “ Ψphysp휉sp훼, 푡q, 휂sp훼, 푡q, 푡q “ 휓s,physp휉sp훼, 푡q, 푡q, 휓bp훼, 푡q “ Ψphysp휉bp훼, 푡q, 휂bp훼, 푡q, 푡q, where 휑s,phys, 휓s,phys represent the values of Φphys and Ψphys on the free surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Following [67] for the periodic case, we assume that there is a background flow of horizontal mean velocity U and the quasi-periodic part of 휑s has the same quasi-periods as 휂s and 휂b (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='29) 휑sp훼, 푡q “ U훼 ` ˜휑sp풌훼, 푡q, ˜휑sp휶, 푡q “ ÿ 풋PZ푑 ˆ휑s 풋p푡q푒푖x풋,휶y, 훼 P R, 휶, 풌 P R푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' According to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2), the bottom boundary is a streamline, therefore 휓b is a constant function (or a function of time only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Considering that adding constants (or functions of time) to Φ and Ψ will not affect the fluid motion, we set ˆ휑s 0 “ 0 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='30) 휓b “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Since Φ and Ψ are harmonic conjugates satisfying boundary conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='29) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='30), we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='31) Φ “ U훼 ` ÿ 풋‰0 ˆ휑풋 cosh ` x풋, 풌yp훽 ` ℎq ˘ cosh ` x풋, 풌yℎ ˘ 푒푖x풋,풌y훼, Ψ “ p훽 ` ℎqU ` ÿ 풋‰0 푖 ˆ휑풋 sinh ` x풋, 풌yp훽 ` ℎq ˘ cosh ` x풋, 풌yℎ ˘ 푒푖x풋,풌y훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Comparing the values of Φ and Ψ at 훽 “ 0 and 훽 “ ´ℎ, we conclude that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='32) 휑sp훼, 푡q “ U훼 ` 퐻cothr휓ssp훼, 푡q, 휓s 훼p훼, 푡q “ 퐻tanhr휑s 훼sp훼, 푡q, 휑b 훼p훼, 푡q “ U ` 퐻sechr휑s 훼sp훼, 푡q “ U ´ 퐻cschr휓s 훼sp훼, 푡q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' QUASI-PERIODIC WATER WAVES OF FINITE DEPTH 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Governing equations in conformal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We now present a derivation of the equations of motion for quasi-periodic surface water waves in a conformal mapping formulation when the fluid is of finite depth and the bottom boundary is not necessarily flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' This is an extension of the results in [74], where the fluid depth is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Since the conformal mapping is time- dependent, even though the bottom profile in physical space is fixed, the width of the strip in the conformal domain and the parameterization of the bottom boundary in conformal space, denoted ℎp푡q and 휁bp훼, 푡q, respectively, both vary with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Therefore besides the free surface, the time evolution equations of ℎ and 휁b in conformal space are needed to describe the evolution of the fluid domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' This is the main difference between the conformal mapping formulations in deep and finite-depth water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' To begin, we use the chain rule to obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='33) 푑푊 푑푤 “ 푑푊phys 푑푧 ¨ 푑푧 푑푤 ñ Φphys 푥 ` 푖Ψphys 푥 “ Φ훼 ` 푖Ψ훼 푥훼 ` 푖푦훼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Since Φphys 푦 “ ´Ψphys 푥 , we can express the velocity of the fluid, which is the gradient of Φphys, in terms of Φ훼 and Ψ훼 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='34) Φphys 푥 “ Φ훼푥훼 ` Ψ훼푦훼 푥2훼 ` 푦2훼 , Φphys 푦 “ Φ훼푦훼 ´ Ψ훼푥훼 푥2훼 ` 푦2훼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Evaluating (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='34) on the free surface, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='35) Φphys 푥 ˇˇˇ 푧“휁sp훼,푡q “ 휑s 훼휉s 훼 ` 휓s 훼휂s 훼 퐽s , Φphys 푦 ˇˇˇ 푧“휁sp훼,푡q “ 휑s 훼휂s 훼 ´ 휓s 훼휉s 훼 퐽s , 퐽s “ p휉s 훼q2 ` p휂s 훼q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Next we derive the kinematic boundary condition in conformal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We define the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='36) 휗 :“ 푧푡 푧푤 “ 푥푡푥훼 ` 푦푡푦훼 푥2훼 ` 푦2훼 ` 푖 푦푡푥훼 ´ 푥푡푦훼 푥2훼 ` 푦2훼 , which is holomorphic on 푆ℎ as long as 푧푤 is bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Evaluating (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='36) at 훽 “ 0 and 훽 “ ´ℎp푡q and replacing the derivatives of 푥 and 푦 by the derivatives of 휉s and 휂s using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='18), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='19), we obtain that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='37) Re 휗 ˇˇˇ 훽“0 “ 휉s 푡휉s 훼 ` 휂s 푡휂s 훼 퐽s , Im 휗 ˇˇˇ 훽“0 “ 휂s 푡휉s 훼 ´ 휉s 푡휂s 훼 퐽s , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='38) Re 휗 ˇˇˇ 훽“´ℎp푡q “ 휉b 푡 휉b 훼 ` 휂b 푡 휂b 훼 퐽b , Im 휗 ˇˇˇ 훽“´ℎp푡q “ 휂b 푡 휉b 훼 ´ 휉b 푡 휂b 훼 퐽b ` ℎ푡, 퐽b “ p휉b 훼q2 ` p휂b 훼q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Furthermore, the substitution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='35) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3) gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='39) 휂s 푡휉s 훼 ´ 휉s 푡휂s 훼 “ ´휓s 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Therefore we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='40) Im 휗 ˇˇˇ 훽“0 “ ´휓s 훼 퐽s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' WILKENING AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ZHAO Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='21) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='38), we obtain that 휂b 푡 휉b 훼 ´ 휉b 푡 휂b 훼 “ 0, thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='41) Im 휗 ˇˇˇ 훽“´ℎp푡q “ ℎ푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Since ℎ푡 does not depend on the spatial variable, similar to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='32), Re 휗|훽“0 and Re 휗|훽“´ℎp푡q can be determined by Im 휗|훽“0 up to an additive constant (that may depend on time but not space) as follows, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='42) 휉s 푡휉s 훼 ` 휂s 푡휂s 훼 퐽s “ ´퐻coth „휓s 훼 퐽s ȷ ` 퐶1, 휉b 푡 휉b 훼 ` 휂b 푡 휂b 훼 퐽b “ 퐻csch „휓s 훼 퐽s ȷ ` 퐶1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Since 휗 is a holomorphic function defined on 푆ℎ, using Cauchy’s integral theorem, we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='43) ż 푎`푖p휖´ℎq ´푎`푖p휖´ℎq ` ż 푎´푖휖 푎`푖p휖´ℎq ` ż ´푎´푖휖 푎´푖휖 ` ż ´푎`푖p휖´ℎq ´푎´푖휖 휗p푤q 푑푤 “ 0, 푎, 휖 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Dividing both sides of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='43) by 2푎 and taking the limit 푎 Ñ 8, 휖 Ñ 0`, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='44) ˆ휗0 “ 푃0r휗p훼qs “ 푃0r휗p훼 ´ 푖ℎqs, where we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='7) in the first equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='40) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='41) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='44), we obtain the the time evolution equation of the width of the strip 푆ℎ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='45) ℎ푡 “ ´푃0 „휓s 훼 퐽s ȷ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Finally, combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='37), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='38) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='42), we obtain the kinematic boundary conditions at both the free surface and the bottom boundary in conformal space (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='46) ˜휉s 푡 휂s 푡 ¸ “ ˜휉s 훼 ´휂s 훼 휂s 훼 휉s 훼 ¸ ¨ ˝´퐻coth ”휓s 훼 퐽s ı ` 퐶1 ´휓s 훼 퐽s ˛ ‚, ˜휉b 푡 휂b 푡 ¸ “ ˜휉b 훼 휂b 훼 ¸ ˆ 퐻csch „휓s 훼 퐽s ȷ ` 퐶1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Since 휉s and 휉b are determined by 휂s and 휂b up to an additive constant 푥0 by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='26), we only need to evolve ℎ, 휂s and 휂b to track the evolution of the fluid domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Comparing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='24) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='46), we know that the free parameter 푥0 is related to 퐶1 through the ODE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='47) 푑푥0 푑푡 “ 푃0 „ 휉s 훼 ˆ ´퐻coth „휓s 훼 퐽s ȷ ` 퐶1 ˙ ` 휂s 훼휓s 훼 퐽s ȷ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Thus, 푥0p푡q is uniquely determined by 퐶1 and 푥0p0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Several choices of 퐶1 have been discussed in detail in [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the scope of this paper, we choose 퐶1 and 푥0p0q as follows (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='48) 퐶1 “ 푃0 “ 휉s 훼퐻cothr휓s 훼{퐽ss ´ 휂s 훼휓s 훼{퐽s‰ , 푥0p0q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' This ensures that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='49) 푥0p푡q “ 0, p푡 ě 0q and alleviates the need to explicitly solve the ODE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Now we derive the dynamic boundary condition at the free surface in conformal space from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Differentiating the first equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='28) with respect to 푡, we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='50) 휑s 푡 “ 휑s,phys 푥 휉s 푡 ` 휑s,phys 푡 , where 휑s,phys 푥 can be expressed in terms of the gradient of Φphys as follows (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='51) 휑s,phys 푥 “ Φphys 푥 ` Φphys 푦 휂s,phys 푥 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' QUASI-PERIODIC WATER WAVES OF FINITE DEPTH 11 From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='33), we know that ˇˇ∇Φphysˇˇ2 “ ´` 휑s 훼 ˘2 ` ` 휓s 훼 ˘2¯ {퐽s at 훽 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Substitution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='46), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='50) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='51) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='4) then gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='52) 휑s 푡 “ ` Φphys 푥 , Φphys 푦 ˘ ˆ휉s 훼 ´휂s 훼 휂s 훼 휉s 훼 ˙ looooooooooooooooomooooooooooooooooon ` 휑s 훼 , ´휓s 훼 ˘ ˆ´퐻coth r휓s 훼{퐽ss ` 퐶1 ´휓s 훼{퐽s ˙ ´ ` 휑s 훼 ˘2 ` ` 휓s 훼 ˘2 2퐽s ´ 푔휂s ` 휏휅 ` 퐶, where 휅 is the mean curvature, given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='53) 휅 “ 휉s 훼휂s 훼훼 ´ 휂s 훼휉s 훼훼 ` 퐽s˘3{2 , and 퐶 is an arbitrary integration constant that may depend on time but not space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the discussion of this paper, we choose 퐶 such that 푃0r휑s 푡s “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In conclusion, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='45), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='46) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='52) are the governing equations in conformal space for finite-depth quasi-periodic gravity-capillary waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Following [74], instead of solving these equations directly, which are posed on the real line, we lift the problem to a higher dimensional torus T푑 and compute the time evolution of the corresponding torus functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' then we evaluate the torus functions along the characteristic direction to obtain quasi-periodic functions on the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Using the torus version of the quasi-periodic derivative and Hilbert transform operators in Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3, we obtain the governing equations on T푑 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='45), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='46) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='52), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='54) ˜휂s 푡 “ ` ´ 퐻cothr ˜휒ss ` 퐶1 ˘ ˜휂s 훼 ´ ˜휉s 훼 ˜휒s, ˜휂b 푡 “ ` 퐻cschr ˜휒ss ` 퐶1 ˘ ˜휂b 훼, ℎ푡 “ ´푃0r ˜휒ss, ˜휑s 푡 “ 푃 „` ˜휓s 훼 ˘2 ´ ` ˜휑s 훼 ` U ˘2 2˜퐽s ` ` 퐶1 ´ 퐻cothr ˜휒ss ˘` ˜휑s 훼 ` U ˘ ´ 푔 ˜휂s ` 휏˜휅 ȷ ˜휉s “ 퐻cothr˜휂ss ` 퐻cschr˜휂bs, ˜휓s “ 퐻tanhr ˜휑ss, ˜퐽s “ ` 1 ` ˜휉s 훼 ˘2 ` ` ˜휂s 훼 ˘2, ˜휒s “ ˜휓s 훼 ˜퐽s , ˜휅 “ ` 1 ` ˜휉s 훼 ˘ ˜휂s 훼훼 ´ ˜휂s 훼 ˜휉s 훼훼 p˜퐽sq3{2 , 퐶1 “ 푃0 “` 1 ` ˜휉s 훼 ˘ 퐻cothr ˜휒ss ´ ˜휂s 훼 ˜휒s‰ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We remark that ˜휑, which is defined on T푑, represents only the quasi-periodic part of 휑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' An extra term U훼 is included in the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='29) to account for the background flow (when present).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Similarly, 휉s and 휉b are obtained from ˜휉s and ˜휉b via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='55) 휉sp훼, 푡q “ 훼 ` ˜휉sp풌훼, 푡q, 휉bp훼, 푡q “ 훼 ` ˜휉bp풌훼, 푡q, where ˜휉s is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='54) and ˜휉b is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='56) ˜휉b “ ´퐻cschr˜휂ss ´ 퐻cothr˜휂bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' According to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='55), we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='57) 휉s 훼p훼, 푡q “ 1 ` ˜휉s 훼p풌훼, 푡q, 휉b 훼p훼, 푡q “ 1 ` ˜휉b 훼p풌훼, 푡q, which is the reason p1 ` ˜휉s 훼q appears in various places in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' WILKENING AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ZHAO Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Modifying the analysis of [74], one can show that if 휁s and 휁b are injective, then 휂s,phys and 휂b,phys are also quasi-periodic functions of the same quasi-periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Moreover, the corresponding torus functions ˜휂s,phys and ˜휂b,phys can be obtained from ˜휂s and ˜휂b by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='58) ˜휂s,physp풙, 푡q “ ˜휂sp풙 ` 풌 ˜Asp풙, 푡q, 푡q, ˜휂b,physp풙, 푡q “ ˜휂bp풙 ` 풌 ˜Abp풙, 푡q, 푡q, where ˜As and ˜Ab satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='59) ˜Asp풙, 푡q ` ˜휉sp풙 ` 풌 ˜Asp풙, 푡q, 푡q “ 0, ˜Abp풙, 푡q ` ˜휉bp풙 ` 풌 ˜Abp풙, 푡q, 푡q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In numerical computations, at any given 푡, one can formulate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='58) as a nonlinear least- squares problem and solve it using a Leveberg-Marquardt method [72], which is discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Since the bottom boundary is stationary, conservation of mass requires that the mean surface height, which we denote by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='60) 휇 “ 1 p2휋q푑 ż T푑 ˜휂phys 푑푥1 ¨ ¨ ¨ 푑푥푑 “ 1 p2휋q푑 ż T푑 ˜휂p1 ` ˜휉훼q 푑훼1 ¨ ¨ ¨ 푑훼푑, is a constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Indeed, one finds that 휇푡 “ 0 by differentiating the second formula of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='60) under the integral sign, integrating the term ˜휂 ˜휉훼푡 by parts with respect to 훼, and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' One usually assumes 휇 “ 0, though for traveling waves it is convenient to first compute the wave assuming ˆ휂0 “ 0 and then adjust ˆ휂0 at the end to achieve 휇 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The governing equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='54) still hold when the bottom boundary of the fluid domain is flat: 휂b,physp푥q “ ´ℎphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the usual case that 휇 “ 0, ℎphys is the mean depth of the fluid in physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Otherwise the mean depth is 휇 ` ℎphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='25), we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='61) 휂b “ ´ℎphys “ ˆ휂s 0 ´ ℎ, which is a constant independent of 훼 and 푡 even though ℎ and ˆ휂s 0 vary in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Moreover, 휉s is related to 휂s by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='62) 휉s “ 훼 ` 퐻cothr휂ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Therefore, when the bottom boundary is flat, one only needs to evolve ˜휂s, ˜휑s and ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Even though we derive (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='54) in the quasi-periodic setting, these equations still hold for the periodic problem if we set 푑 “ 1 and 풌 “ p1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' To obtain the governing equations on T, one just needs to replace the quasi-periodic Hilbert transforms by their periodic counterparts in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='54): (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='63) 퐻tanhr 푓 sp훼q “ ÿ 푗‰0 푖 tanhp푗ℎq ˆ푓푗푒푖푗훼, 퐻cothr 푓 sp훼q “ ÿ 푗‰0 p´푖q cothp푗ℎq ˆ푓푗푒푖푗훼, 퐻cschr 푓 sp훼q “ ÿ 푗‰0 푖 cschp푗ℎq ˆ푓푗푒푖푗훼, 퐻sechr 푓 sp훼q “ ÿ 푗‰0 sechp푗ℎq ˆ푓푗푒푖푗훼, where 푓 is defined on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' If 푑 ą 1, the periodic problem may be embedded in the quasi-periodic problem by assuming that each of the torus functions in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='54) is independent of 훼2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' , 훼푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We conclude this section by explaining how to compute the trajectories of fluid particles in the conformal mapping formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We denote the trajectory of a fluid particle by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='64) 푧푝p푡q “ 푥푝p푡q ` 푖푦푝p푡q “ 푥p훼푝p푡q, 훽푝p푡q, 푡q ` 푖푦p훼푝p푡q, 훽푝p푡q, 푡q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Unlike physical space, where the particle trajectory is computed through (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='65) 9푥푝 “ Φphys 푥 p푥푝p푡q, 푦푝p푡q, 푡q, 9푦푝 “ Φphys 푦 p푥푝p푡q, 푦푝p푡q, 푡q, QUASI-PERIODIC WATER WAVES OF FINITE DEPTH 13 in conformal space, one needs to compute the time evolution of 훼푝p푡q and 훽푝p푡q since the conformal mapping is time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' To do so, we differentiate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='64) and use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='65) to obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='66) 9푧푝 “ Φphys 푥 ` 푖Φphys 푦 “ ` 푥훼 9훼푝 ´ 푦훼 9훽푝 ` 푥푡 ˘ ` 푖 ` 푦훼 9훼푝 ` 푥훼 9훽푝 ` 푦푡 ˘ , where we use the fact that 푥 and 푦 are harmonic conjugates in the last equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Therefore we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='67) ˜ 9훼푝 9훽푝 ¸ “ 1 푥2훼 ` 푦2훼 ˜ 푥훼 푦훼 ´푦훼 푥훼 ¸ ˜ Φphys 푥 ´ 푥푡 Φphys 푦 ´ 푦푡 ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='34) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='36) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='67), we obtain the time evolution equation of 훼푝 and 훽푝 as follows (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='68) ˜ 9훼푝 9훽푝 ¸ “ 1 푥2훼 ` 푦2훼 ˜ Φ훼 ´Ψ훼 ¸ ´ ˜ Rep푧푡{푧푤q Imp푧푡{푧푤q ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' On the free surface, according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='40) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='42), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='68) reads (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='69) ˜ 9훼푝 9훽푝 ¸ “ 1 퐽푠 ˜ 휑s 훼 ´휓s 훼 ¸ ` ˜ 퐻cothr휓s 훼{퐽푠s ´ 퐶1 휓s 훼{퐽푠 ¸ “ ˜ 휑s 훼{퐽푠 ` 퐻cothr휓s 훼{퐽푠s ´ 퐶1 0 ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Thus, 훽푝p푡q “ 훽푝p0q “ 0 and we only need to evolve 훼푝 to track the particle trajectory on the free surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Quasi-periodic traveling waves 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Governing equations of quasi-periodic traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For traveling waves, the system should be translation invariant, so we assume the bottom boundary is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' According to Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='6, we only need to consider the surface variables in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Thus, to simplify the notation, we drop the superscript “s” in these variables in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Moreover, as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1, we focus our discussion on the laboratory frame and assume that there is no background flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Since the bottom boundary is flat, 휉, 휂 and 휑, 휓 are related by Hilbert transforms (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1) 휉 “ 훼 ` 퐻cothr휂s, 휉훼 “ 1 ` 퐻cothr휂훼s, 휑 “ 퐻cothr휓s, 휑훼 “ 퐻cothr휓훼s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We assume the wave is traveling from left to right at speed 푐;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' therefore, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2) 휂physp푥, 푡q “ 휂phys 0 p푥 ´ 푐푡q, 휑physp푥, 푡q “ 휑phys 0 p푥 ´ 푐푡q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Differentiating both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2) with respect to 푥 and 푡 separately, we know that a traveling solution satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3) 휂phys 푡 “ ´푐휂phys 푥 , 휑phys 푡 “ ´푐휑phys 푥 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Substituting the second equation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='20) into the first equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3) and multiplying both sides of the equation by 휉s 훼, we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='4) 휂푡휉훼 ´ 휉푡휂훼 “ ´푐휂훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Comparing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='39), we conclude that a traveling solution satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5) 휓훼 “ 푐휂훼 in conformal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Applying the Hilbert transform 퐻coth to both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='6) 휑훼 “ 푐p휉훼 ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' WILKENING AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ZHAO Substituting the traveling condition of 휑phys in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='50) and employing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='51) to express 휑phys 푥 in terms of the gradient of Φphys, we obtain that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='7) 휑푡 “ ` Φphys 푥 ` Φphys 푦 휂phys 푥 ˘ p휉푡 ´ 푐q “ 휑훼 휉훼 ` 휉훼 ` ´ 퐻coth”휓훼 퐽 ı ` 퐶1 ˘ ` 휂훼 휓훼 퐽 ´ 푐 ˘ “ 휑훼 휉훼 ˆ 휉훼 ` ´ 퐻coth”휓훼 퐽 ı ` 퐶1 ˘ ` 푐 ` 휂훼 ˘2 퐽 ´ 푐 ˙ “ 휑훼 ˆ ´ 퐻coth”휓훼 퐽 ı ` 퐶1 ´ 푐휉훼 퐽 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Here in the second equality, we use the first equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='20) to rewrite 휂phys 푥 as 휂s 훼{휉s 훼 and substitute the gradient of Φphys and 휉푡 using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='35) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='46), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the third equality, we use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5) to replace 휓훼 by 푐휂훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The substitution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='7) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='52) gives (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='8) 푐 퐽 ` 휑훼휉훼 ` 휓훼휂훼 ˘ ´ 1 2퐽 ` p휑훼q2 ` p휓훼q2˘ ´ 푔휂 ` 휏휅 ` 퐶 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='6) to express 휑훼 and 휓훼 in terms of 휉훼 and 휂훼, respectively, we obtain the governing equation of traveling waves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='9) 푃 „ 푐2 2퐽 ` 푔휂 ´ 휏휅 ȷ “ 0, where we choose the integration constant 퐶 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='8) such that 푃0 acting on the left-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='8) returns zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Since (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='9) does not depend on time, the solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='9) can be considered as the initial condition of a traveling wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='53), we know that 퐽 and 휅 are determined by 휂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' hence, the unknowns in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='9) are 휏, 푐 and 휂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Even though we are mainly interested in the case where 휂 is quasi-periodic, the governing equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='9) still holds when 휂 is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Due to the projection operator, modifying 휂 by a constant will not influence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' hence, we assume that 푃0r휂s “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In this paper, we focus on traveling waves with even symmetry (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='10) 휂p훼q “ 휂p´훼q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We compute 휉 from 휂 using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1) and deduce that 휉 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Asymmetric traveling waves have been studied in [31,68,79] in the periodic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' As in the initial value problem, we first solve for ˜휂 on T푑 and then reconstruct 휂 from ˜휂 using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The governing equations of traveling waves on the torus read (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11) Rr휏, 푏, ˜휂s “ 푃 „ 푏 2˜퐽 ` 푔 ˜휂 ´ 휏˜휅 ȷ “ 0, ˜휉 “ 퐻cothr˜휂s, ˜퐽 “ ` 1 ` ˜휉훼 ˘2 ` ˜휂2 훼, ˜휅 “ ` 1 ` ˜휉훼 ˘ ˜휂훼훼 ´ ˜휂훼 ˜휉훼훼 ˜퐽3{2 , where 푏 “ 푐2 and R is called the residual function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We treat the strip width ℎ in conformal space as a fixed parameter and suppress it in the argument list of R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Linearizing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11) around the zero solution ˜휂 “ 0, we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='12) 푏퐻cothr훿 ˜휂훼s ´ 푔훿 ˜휂 ` 휏훿 ˜휂훼훼 “ 0, QUASI-PERIODIC WATER WAVES OF FINITE DEPTH 15 where 훿 ˜휂 denotes the variation of ˜휂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Expressing 훿 ˜휂 in terms of its Fourier series in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='12), we obtain the dispersion relation for the linearized problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='13) 푏 cothpx풋, 풌yℎqx풋, 풌y ´ 푔 ´ 휏px풋, 풌yq2 “ 0, 풋 P 푍푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Since the entries of 풌 are linearly independent over Z, given 푏 and 휏, there exist at most two linearly independent vectors 풋1, 풋2 P Z푑 that satisfy the dispersion relation [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For simplicity, we consider the basic case where 푑 “ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' hence, 휂 possesses two quasi-periods and ˜휂 is defined on T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Without loss of generality, we also assume that 풋1 “ p1, 0q푇, 풋2 “ p0, 1q푇 and 풌 “ p1, 푘q푇, where 푘 is a positive irrational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In summary, we study quasi-periodic traveling waves of the following form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='14) 휂p훼q “ ˜휂p훼, 푘훼q, ˜휂p훼1, 훼2q “ ÿ 푗1,푗2PZ ˆ휂푗1,푗2푒푖p푗1훼1`푗2훼2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We also assume that ˜휂 is an even function with zero mean on T2, which is consistent with the assumptions on 휂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Therefore the Fourier coefficients of ˜휂 satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='15) ˆ휂0,0 “ 0, ˆ휂푗1,푗2 “ ˆ휂´푗1,´푗2 P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Here ˜휂 has zero mean in conformal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We refer to Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1 below if one wants to obtain solutions with zero mean in physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Under assumptions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='15), we can study the problem of quasi-periodic traveling waves in the setting of a bifurcation problem with a two-dimensional kernel spanned by the solutions of the linearized problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='12): (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='16) ˜휂linp훼1, 훼2q “ ˆ휂1,0p푒푖훼1 ` 푒´푖훼1q ` ˆ휂0,1p푒푖훼2 ` 푒´푖훼2q, 푏lin “ 푐2 lin “ 푔p푘2 ´ 1q 푘p푘 cothpℎq ´ cothp푘ℎqq, 휏lin “ 푔p푘 cothp푘ℎq ´ cothpℎqq 푘p푘 cothpℎq ´ cothp푘ℎqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We refer to ˆ휂1,0 and ˆ휂0,1 as the base Fourier coefficients and the corresponding Fourier modes 푒˘푖훼1, 푒˘푖훼2 as the base Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Nonlinear solutions can be considered as bifurcations from the zero-amplitude solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We usually choose the base Fourier coefficients as bifur- cation parameters and fix them at nonzero values to ensure that the solutions we obtain are genuinely quasi-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In finite depth, ℎ is a third parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' As shown in [75], large-amplitude quasi-periodic traveling solutions can often be found by searching for secondary bifurcations from finite-amplitude periodic traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The linearization of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11) around a periodic solution reads (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='17) 훿R “ 푃 « 훿푏 2˜퐽 ´ 1 2˜퐽2 푏훿˜퐽 ` 푔훿 ˜휂 ´ 훿휏˜휅 ´ 휏훿 ˜휅 ff , 훿 ˜휉훼 “ 퐻cothr훿 ˜휂훼s, 훿˜퐽 “ 2 ´ p1 ` ˜휉훼q훿 ˜휉훼 ` ˜휂훼훿 ˜휂훼 ¯ , 훿 ˜휅 “ ´3 2 ˜휅 훿˜퐽 ˜퐽 ` 1 ˜퐽3{2 ´ 훿 ˜휉훼 ˜휂훼훼 ` p1 ` ˜휉훼q훿 ˜휂훼훼 ´ 훿 ˜휂훼 ˜휉훼훼 ´ ˜휂훼훿 ˜휉훼훼 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Let 푞 denote the triple p휏, 푏, ˜휂q and let 픮perp푠q denote a one-parameter family of periodic trav- eling waves embedded in the quasi-periodic framework by assuming ˜휂p훼1, 훼2q is independent of 훼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Here 푠 is an amplitude parameter (such as ˆ휂1,0), and, for simplicity, we fix 휏 and the strip width ℎ in conformal space to be independent of 푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Each solution 푞 “ 픮perp푠q in the family satisfies R ` 푞 ˘ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In [75], an algorithm is presented for locating bifurcation points by using a 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' WILKENING AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ZHAO quadratically convergent root bracketing technique [13] to locate zeros of the signed smallest singular value (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='18) 휒p푠q “ sgn ´ det ´ J quap푠q ¯¯ 휎min ´ J quap푠q ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Here J quap푠q is a Fourier truncation of the restricted Jacobian obtained from the linearization (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='17) applied only in quasi-periodic perturbation directions of the form 훿푞 “ p0, 0, 훿 ˜휂quaq, where 훿 ˜휂qua has 2D Fourier modes x 훿 ˜휂qua 푗1,푗2 that are all zero unless 푗2 P t1, ´1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' This construction is based on Bloch-Fourier perturbation theory over periodic potentials [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' At zeros of 휒p푠q, J quap푠q has a kernel that provides a bifurcation direction 훿 ˜휂qua that allows us to switch from the primary periodic branch to the secondary quasi-periodic branch of traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We use ˜휂per ` 휖훿 ˜휂qua, with 휖 chosen empirically, as an initial guess for solutions on this secondary branch, and then use numerical continuation to follow the branch beyond the realm of linearization about the primary branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Further discussion of the analysis and computation of the bifurcation problem in the infinite-depth setting is given in [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We have simplified the computation of quasi-periodic traveling waves via the conformal mapping formulation by setting ˆ휂0,0 “ 0 and fixing the strip width ℎ in conformal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The mean surface height and depth of the bottom boundary in physical space, 휇 and ℎphys, can then be computed from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='60) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='61), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' If desired, after computing a solution with ˆ휂0,0 “ 0, one can adjust the height of the traveling wave by setting ˆ휂0,0 “ ´휇 and 휇 “ 0, assigned in that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The resulting solution satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='19) ˆ휂0,0 “ ´푃0r ` 푃r˜휂s ˘ p1 ` ˜휉훼qs, where 푃r˜휂s on the right-hand side is the initially computed solution with ˆ휂0,0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Another option is to prescribe 휇 “ 0, ℎphys, ˆ휂1,0 and ˆ휂0,1 and solve for ˆ휂0,0 and ℎ along with the remaining Fourier modes ˆ휂푗1,푗2 using the Levenberg-Marquardt solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' This would entail including ℎ “ ℎphys ` ˆ휂0,0 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='61) as well as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='19) as additional constraints in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Weakly nonlinear approximations of quasi-periodic traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Although the primary focus of this work is on computing quasi-periodic solutions of the fully nonlinear time-dependent and traveling water wave equations in finite depth, it is instructive to in- vestigate how small divisors arise in weakly nonlinear approximations of small-amplitude quasi-periodic traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In previous work, it has been necessary to treat such small divisors carefully using Nash-Moser theory [37, 53] to prove existence of temporally quasi- periodic water waves [7–10,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Here we focus on spatial quasi-periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1, the traveling solutions bifurcating from the zero solution form a three-parameter family with bifurcation parameters ˆ휂1,0, ˆ휂0,1 and ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the weakly nonlinear model, we treat ℎ as a constant and set these two Fourier coefficients to be fixed, non-zero multiples of an amplitude parameter 휖 and aim to express 푏, 휏 and the other Fourier coefficients of ˜휂 in terms of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Let us consider the following asymptotic expansions of 푏, 휏 and ˜휂 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='20) 푏 “푏p0q ` 휖푏p1q ` 휖2푏p2q ` 휖3푏p3q ` 푂p휖4q, 휏 “휏p0q ` 휖휏p1q ` 휖2휏p2q ` 휖3휏p3q ` 푂p휖4q, ˜휂 “휖 ˜휂p1q ` 휖2 ˜휂p2q ` 휖3 ˜휂p3q ` 푂p휖4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' QUASI-PERIODIC WATER WAVES OF FINITE DEPTH 17 Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='20) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11) and eliminating the coefficients of 휖푛 for 푛 “ 0, 1, 2, we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='21) 푂p1q : 푃 „1 2푏p0q ȷ “ 0, 푂p휖q : 푃 „1 2푏p1q ` 푔 ˜휂p1q ´ 푏p0q퐻coth“ ˜휂p1q 훼 ‰ ´ 휏p0q ˜휂p1q 훼훼 ȷ “ 0, 푂p휖2q : 푃 „1 2푏p2q ` 푔 ˜휂p2q ´ 푏p0q퐻coth“ ˜휂p2q 훼 ‰ ´ 휏p0q ˜휂p2q 훼훼 ´ 푏p1q퐻coth“ ˜휂p1q 훼 ‰ ´ 휏p1q ˜휂p1q 훼훼 ` 푏p0q ˆ3 2 ´ 퐻coth“ ˜휂p1q 훼 ‰¯2 ´ 1 2 ` ˜휂p1q 훼 ˘2 ˙ ` 휏p0q ´ 2퐻coth“ ˜휂p1q 훼 ‰ ˜휂p1q 훼훼 ` 퐻coth“ ˜휂p1q 훼훼 ‰ ˜휂p1q 훼 ¯ ȷ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Since the constant term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='21) vanishes under the projection, we rewrite the second equation as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='22) 푃 „ 푏p0q퐻coth“ ˜휂p1q 훼 ‰ ´ 푔 ˜휂p1q ` 휏p0q ˜휂p1q 훼훼 ȷ “ 0, which is the same as the linearization (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' therefore, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='23) ˜휂p1q “ ˜휂lin “ ˆ휂1,0푒푖훼1 ` ˆ휂0,1푒푖훼2 ` 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=', 푏p0q “ 푏lin, 휏p0q “ 휏lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Using the property of the projection operator and the assumption that 푃0r˜휂s “ 0, we rewrite the third equation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='21) as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='24) 푔 ˜휂p2q ´ 푏p0q퐻coth“ ˜휂p2q 훼 ‰ ´ 휏p0q ˜휂p2q 훼훼 loooooooooooooooooooomoooooooooooooooooooon 퐴p2q ´푏p1q퐻coth“ ˜휂p1q 훼 ‰ ´ 휏p1q ˜휂p1q 훼훼 loooooooooooooooomoooooooooooooooon 퐵p2q “ 푃 „ 푏p0q ˆ ´3 2 ´ 퐻coth“ ˜휂p1q 훼 ‰¯2 ` 1 2 ` ˜휂p1q 훼 ˘2 ˙ ´ 휏p0q ´ 2퐻coth“ ˜휂p1q 훼 ‰ ˜휂p1q 훼훼 ` 퐻coth“ ˜휂p1q 훼훼 ‰ ˜휂p1q 훼 ¯ȷ looooooooooooooooooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooooooooooooooooon 퐶p2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Substituting ˜휂p1q, ˜푏p0q and ˜휏p0q into 퐶p2q using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='23), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='25) 퐶p2q “ ˆ퐶p2q 2,0푒푖p2훼1q ` ˆ퐶p2q 0,2푒푖p2훼2q ` ˆ퐶p2q 1,1푒푖p훼1`훼2q ` ˆ퐶p2q 1,´1푒푖p훼1´훼2q ` 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=', where the Fourier coefficients of 퐶p2q are (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='26) ˆ퐶p2q 2,0 “ 푔 ˆ휂2 1,0 3p푘2 ` 1q coth2pℎq ´ 6푘 cothp푘ℎq cothpℎq ` 푘2 ´ 1 2푘pcothp푘ℎq ´ 푘 cothpℎqq , ˆ퐶p2q 0,2 “ ´푔푘 ˆ휂2 0,1 3p푘2 ` 1q coth2p푘ℎq ´ 6푘 cothp푘ℎq cothpℎq ´ 푘2 ` 1 2pcothp푘ℎq ´ 푘 cothpℎqq , ˆ퐶p2q 1,1 “ ´푔 ˆ휂1,0 ˆ휂0,1 p푘2 ` 2푘q coth2p푘ℎq ´ p2푘 ` 1q coth2pℎq ` p´푘2 ` 1q cothp푘ℎq cothpℎq ´ 푘2 ` 1 cothp푘ℎq ´ 푘 cothpℎq , ˆ퐶p2q 1,´1 “ 푔 ˆ휂1,0 ˆ휂0,1 p푘2 ´ 2푘q coth2p푘ℎq ` p2푘 ´ 1q coth2pℎq ` p푘2 ´ 1q cothp푘ℎq cothpℎq ´ 푘2 ` 1 cothp푘ℎq ´ 푘 cothpℎq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 18 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' WILKENING AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ZHAO We observe that 퐴p2q is linear with respect to ˜휂p2q and the Fourier coefficients of 퐴p2q can be expressed as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='27) ˆ퐴p2q 푗1,푗2 “ ˆ푆푗1,푗2 ˆ휂p2q 푗1,푗2, where the symbol ˆ푆푗1,푗2 is defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='28) ˆ푆푗1,푗2 “ 푔 ´ 푏p0q cothpp푗1 ` 푘푗2qℎqp푗1 ` 푘푗2q ` 휏p0qp푗1 ` 푘푗2q2 “ 푔 푘 ˆ 푘 ` 푘2 ´ 1 cothp푘ℎq ´ 푘 cothpℎq cothpp푗1 ` 푘푗2qℎqp푗1 ` 푘푗2q ` cothpℎq ´ 푘 cothp푘ℎq cothp푘ℎq ´ 푘 cothpℎqp푗1 ` 푘푗2q2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Since ˆ푆˘1,0 and ˆ푆0,˘1 are both zero according to the definition, we know that ˆ퐴p2q ˘1,0 “ ˆ퐴p2q 0,˘1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We also observe that 퐵p2q is linear with respect to ˜휂p1q with Fourier coefficients (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='29) ˆ퐵p2q 푗1,푗2 “ ˆ푄p1q 푗1,푗2 ˆ휂p1q 푗1,푗2, ˆ푄p푛q 푗1,푗2 “ ´푏p푛q cothpp푗1 ` 푘푗2qℎqp푗1 ` 푘푗2q ` 휏p푛qp푗1 ` 푘푗2q2, where p푗1, 푗2q “ p˘1, 0q, p0, ˘1q according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='26), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='27) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='29), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='30) 푏p1q “ 휏p1q “ 0, ˆ휂p2q 푗1,푗2 “ $ ’ & ’ % 퐶p2q 푗1,푗2 ˆ푆푗1,푗2 , |푗1| ` |푗2| “ 2, 0, |푗1| ` |푗2| ‰ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' One can obtain the asymptotic expansions of quasi-periodic traveling waves in the case of deep water by letting ℎ go to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In this case, the expressions of ˜휂p1q, 푏p0q and 휏p0q read (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='31) ˜휂p1q “ ˆ휂1,0푒푖훼1 ` ˆ휂0,1푒푖훼2 ` 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=', 푏p0q “ 푔 ` 푔 푘 , 휏p0q “ 푔 푘 and the expressions of ˜휂p2q, 푏p1q and 휏p1q read (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='32) ˜휂p2q “ ˆ휂p2q 2,0푒푖p2훼1q ` ˆ휂p2q 0,2푒푖p2훼2q ` ˆ휂p2q 1,1푒푖p훼1`훼2q ` ˆ휂p2q 1,´1푒푖p훼1´훼2q ` 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=', 푏p1q “ 휏p1q “ 0, ˆ휂p2q 2,0 “ ´ ˆ휂2 1,0푔p2푘 ´ 1q{푘 ˆ푆2,0 , ˆ휂p2q 0,2 “ ˆ휂2 0,1푔푘p푘 ´ 2q ˆ푆0,2 , ˆ휂p2q 1,1 “ ´ ˆ휂1,0 ˆ휂0,1푔p푘 ` 1q ˆ푆1,1 , ˆ휂p2q 1,´1 “ ´ ˆ휂1,0 ˆ휂0,1푔p푘 ` 1q ˆ푆1,´1 , where (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='33) ˆ푆푗1,푗2 “ 푔 푘 p|푗1 ` 푘푗2| ´ 푘qp|푗1 ` 푘푗2| ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Even though we stop at the second order in the weakly nonlinear model, one can continue computing higher-order terms by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Suppose that we have obtained terms of order 푛 ´ 1 for ˜휂 and terms of order 푛 ´ 2 for 푏 and 휏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Eliminating the coefficients of 휖푛 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11), we find that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='34) 푔 ˜휂p푛q ´ 푏p0q퐻coth“ ˜휂p푛q 훼 ‰ ´ 휏p0q ˜휂p푛q 훼훼 ´ 푏p푛´1q퐻coth“ ˜휂p1q 훼 ‰ ´ 휏p푛´1q ˜휂p1q 훼훼 “ 퐶p푛q, where 퐶p푛q depends on ␣ 푏p푗q( 0ď푗ď푛´2, ␣ 휏p푗q( 0ď푗ď푛´2 and ␣ ˜휂p푗q( 0ď푗ď푛´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Comparing the Fourier coefficients of both sides of the above equation, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='35) ˆ푆푗1,푗2 ˆ휂p푛q 푗1,푗2 ` ˆ푄p푛´1q 푗1,푗2 ˆ휂p1q 푗1,푗2 “ ˆ퐶p푛q 푗1,푗2, QUASI-PERIODIC WATER WAVES OF FINITE DEPTH 19 where ˆ푆푗1,푗2 and ˆ푄p푛´1q 푗1,푗2 are given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='28) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='29), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Eventually we can express 푏p푛´1q, 휏p푛´1q and the Fourier coefficients of ˜휂p푛q as follows, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='36) ˆ휂p푛q 푗1,푗2 “ ˆ퐶p푛q 푗1,푗2 ˆ푆푗1,푗2 , p푗1, 푗2q ‰ p˘1, 0q, p0, ˘1q, 푏p푛´1q “ ˆ퐶p푛q 0,1 ˆ휂0,1 ´ 푘2 ˆ퐶p푛q 1,0 ˆ휂1,0 푘p푘 cothpℎq ´ cothp푘ℎqq, 휏p푛´1q “ cothpℎq ˆ퐶p푛q 0,1 ˆ휂0,1 ´ 푘 cothp푘ℎq ˆ퐶p푛q 1,0 ˆ휂1,0 푘p푘 cothpℎq ´ cothp푘ℎqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Note that the Fourier coefficients of ˜휂p푛q are obtained through a division by ˆ푆푗1,푗2 for p푗1, 푗2q ‰ p˘1, 0q, p0, ˘1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' If the ˆ푆푗1,푗2 can become arbitrarily small, the corresponding terms ˆ휂p푛q 푗1,푗2 may be strongly amplified, calling into question the nature of the expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' This is known as a small divisor problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the case of deep water, it is clear from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='33) that some of the ˆ푆푗1,푗2 approach zero as |푗1|, |푗2| grow without bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Speculating on the possibilities, it may be that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='20) becomes aysmptotic series provided that 푘 is sufficiently irrational, satisfying a diophantine condition [46] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='37) |푘 ´ 푗1{푗2| ą 퐶|푗2|´휈, 푗1 P Z, 푗2 P Zzt0u, where 퐶 is a positive constant and 휈 ą 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' But it may also be that exact mathematical solutions only exist for sufficiently small values of 휖 in a totally disconnected Cantor-like set [37], even under the assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' More research is needed to resolve these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The story is even more complicated in the case where the fluid is of finite depth because the expression for ˆ푆푗1,푗2 involves the hyperbolic cotangent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' But this formula becomes simpler again in the case of shallow water, where ℎ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Expanding cothpℎq and cothp푘ℎq in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='28) in a Laurent expansion about ℎ “ 0, we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='38) ˆ푆푗1,푗2 “ 푔ℎ4 45 p|푗1 ` 푘푗2|2 ´ 푘2qp|푗1 ` 푘푗2|2 ´ 1q ` 푂pℎ6q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We notice that ˆ푆푗1,푗2 can be very small due to the factor of ℎ4 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Thus, in the shallow water regime, the amplitudes of quasi-periodic traveling waves bifurcating from the zero-amplitude solution must be small, with 휖 at most 푂pℎ4q, if weakly nonlinear theory is to predict their behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Numerical methods and results As in Section 3 above, we focus our discussion on quasi-periodic functions with two quasi- periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' All computation will be performed with respect to torus functions on T2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' the one- dimensional quasi-periodic functions will be reconstructed from the torus functions using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Let 푓 p훼q be a quasi-periodic function with two quasi-periods and let ˜푓 denote the corresponding periodic function on T2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1) 푓 p훼q “ ˜푓 p훼, 푘훼q, ˜푓 p훼1, 훼2q “ ÿ 푗1,푗2PZ ˆ푓푗1,푗2푒푖p푗1훼1`푗2훼2q, p훼1, 훼2q P T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Following [73,74], we adopt a pseudo-spectral method and represent ˜푓 in two ways: (1) Via the values of ˜푓 on a uniform 푀1 ˆ 푀2 grid on the torus T2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2) ˜푓푚1,푚2 “ ˜푓 p2휋푚1{푀1 , 2휋푚2{푀2q, 0 ď 푚1 ă 푀1 , 0 ď 푚2 ă 푀2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' WILKENING AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ZHAO (2) Via the truncated two-dimensional Fourier series of ˜푓 , with Fourier coefficients given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3) ˆ푓푗1,푗2 “ 1 푀2 푀2´1 ÿ 푚2“0 ˜ 1 푀1 푀1´1 ÿ 푚1“0 ˜푓푚1,푚2푒´2휋푖푗1푚1{푀1 ¸ 푒´2휋푖푗2푚2{푀2, 0 ď 푗1 ď 푀1{2, ´푀2{2 ă 푗2 ď 푀2{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We use the ‘r2c’ and ’c2r’ version of the 2d FFTW library to rapidly transform between these two forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Products, powers and quotients in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='54) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11) are evaluated point-wise on the grid while derivatives and Hilbert transforms are computed in Fourier space via Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the scope of this paper, we choose 푘 “ 1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 2 for all numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Time evolution of spatially quasi-periodic waves of finite depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' To compute the time evolution of spatially quasi-periodic waves, we discretize (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='54) on T2 and use the fifth-order explicit Runge-Kutta method of Dormand and Prince [32, 74] as the time stepping scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The initial condition of the water wave is given in physical space, which is more natural in practice, and we compute the conformal mapping to transform the initial condition from physical space to conformal space using the method described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The numerical examples discussed below are gravity waves but our numerical method also applies to the case of nonzero surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Time evolution of an initially flat free surface in the presence of a background flow and a quasi-periodic bottom boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Figure 2 shows the time evolution of a free surface wave that is initially flat and develops quasi-periodic dynamics in the presence of a background flow and a quasi-periodic bottom boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In physical space, the bottom boundary is parameterized by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='4) 휂b,physp푥q “ ´1 ` 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2 cosp푥q ` 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2 cosp푥{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 2q and the mean velocity of the background flow in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='29) is U “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the computation, we use 푀1 “ 푀2 “ 512 and compute the time evolution of the wave from 푡 “ 0 to 푡 “ 3 with time steps Δ푡 “ 10´5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In panel (a), the black line plots the bottom boundary and the blue line plots the flat free surface at 푡 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' To better distinguish the shape of the free surface at QUASI-PERIODIC WATER WAVES OF FINITE DEPTH 21 different times, we plot the free surface at different times with an upward spatial shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The time difference between two adjacent curves is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='06, and we plot (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5) 휂sp훼, 푡푛q ` 10푡푛{3, 푡푛 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='06푛, 푛 “ 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' , 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Due to the background flow and the quasi-periodic bottom boundary, the free surface wave moves from left to right and forms wave crests ahead of the peaks of the bottom boundary, which deflects the fluid upward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Panels (b) and (c) show snapshots of the time evolution of the free surface from 푡 “ 0 to 푡 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5 and from 푡 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5 to 푡 “ 3 separately without the upward shift given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' the time difference between two adjacent curves in both panels is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' One can observe that the free surface gradually develops quasi-periodic crests and troughs, which drift from left to right due to the background flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' the height difference between crests and neighboring troughs increases with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Figure 3 shows the time evolution of an initially periodic free surface wave in the presence of a periodic bottom boundary whose spatial period is irrationally related to the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In physical space, the initial free surface and the bottom boundary are given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='6) 휂s,phys 0 p푥q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2 cosp푥q, 휂b,physp푥q “ ´1 ` 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2 cosp푥{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In panel (a), the initial free surface and the bottom boundary are plotted with blue and black curves, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' As shown in the figure, they are both periodic and the bottom boundary’s wavelength is longer than that of the free surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We use 푀1 “ 푀2 “ 256 in the computation and evolve the water wave from 푡 “ 0 to 푡 “ 10 with time steps Δ푡 “ 2 ˆ 10´5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' At 푡 “ 0, the fluid is at rest with zero velocity potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In panel (a), we plot the time evolution of the free surface with an upward spatial shift (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='7) 휂sp훼, 푡푛q ` 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='75푡푛, 푡푛 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2푛, 푛 “ 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' , 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The free surface flattens due to the force of gravity and rises again due to inertia, which is similar to the oscillation of a standing water wave [44, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' One can observe that the crests and troughs of the surface wave are not symmetric for 푡 ą 0 except at 푥 “ 0 due to the even symmetry of the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In panels (b), (c) and (d), we plot snapshots of the time evolution of the free surface without the upward shit (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='7) from 푡 “ 0 to 푡 “ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' from 푡 “ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Time evolution of an initially periodic free surface in the presence of a periodic bottom boundary whose spatial period is irrationally related to the period of the free surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' WILKENING AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ZHAO Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Panel (a) shows the velocity field of the fluid corresponding to the free surface wave in Figure 2 at 푡 “ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Panel (b) shows the velocity field of the fluid corresponding to the wave in Figure 3 at 푡 “ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The colors represent the magnitude of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' to 푡 “ 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' and from 푡 “ 7 to 푡 “ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The time difference between two adjacent curves is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' One can observe that the wave oscillates up and down like a standing wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' However, as a consequence of the quasi-periodic interactions between the surface wave and the bottom boundary, the heights of different crests are different at any given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In Figure 4, we plot the velocity field of the waves in Figures 2 and 3 at the final times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The arrows denote the direction of the velocity field and the colors represent the magnitude of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Panel (a) corresponds to the free surface wave in Figure 2 at 푡 “ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' One can observe that at any point in the fluid, the velocity field’s horizontal component is positive: Φphys 푥 ą 0, which is consistent with the presence of a background flow from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' At the bottom boundary, the velocity field is parallel to the boundary, which satisfies the Neumann boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Panel (b) shows the velocity field of the fluid corresponding to the wave in Figure 3 at 푡 “ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Unlike panel (a), since there is no background flow, the direction of the velocity field’s horizontal component varies throughout the fluid, and there are points where the horizontal component vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The velocity magnitude is relatively large in the yellow jets, where one can deduce from the direction of the velocity field that crests are forming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Spatially quasi-periodic traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We formulate the traveling wave problem as a nonlinear least-squares problem, which we solve using a variant of the Levenberg-Marquardt algorithm [50, 72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1, we introduced the residual function R in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11), which depends on 휏, 푏, ˜휂, and demonstrated that the solutions of the traveling wave problem are the solutions of Rr휏, 푏, ˜휂s “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the computation, we consider 휏, 푏 and the Fourier coefficients of ˜휂 as unknowns, denoted ˆ휂, and define the following scalar objective function (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='8) Fr휏, 푏, ˆ휂s :“ 1 8휋2 ż T2 R2r휏, 푏, ˆ휂s 푑훼1 푑훼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 0 个个 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5 个 1 2元 2 元 4π 6T 8元 10元 012 10 8 6 4 20 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5 个 → 个 2元 0 2 元 4π 6元 8元 10元0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2 0QUASI-PERIODIC WATER WAVES OF FINITE DEPTH 23 Note that solving (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11) is equivalent to finding a zero of the objective function Fr휏, 푏, ˆ휂s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For the unknown ˆ휂, we only vary the leading Fourier coefficients ˆ휂푗1,푗2 with |푗1| ď 푁1 ă 푀1{2, |푗2| ď 푁2 ă 푀2{2 and set the other Fourier coefficients to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' According to the assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='15), we also set ˆ휂0,0 “ 0 and require that the Fourier coefficients ˆ휂푗1,푗2 are real and satisfy ˆ휂´푗1,´푗2 “ ˆ휂푗1,푗2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Consequently, the number of independent leading Fourier coefficients is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='9) 푁tot “ 푁1p2푁2 ` 1q ` 푁2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1, we choose ˆ휂1,0, ˆ휂0,1 and ℎ as bifurcation parameters when com- puting quasi-periodic traveling solutions bifurcating from the zero-amplitude solution and fix them at nonzero amplitudes in the minimization of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Therefore there are 푁tot parameters to compute, which are stored in a vector 풑 as follows (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='10) 푝1 “ 휏, 푝2 “ ˆ휂1,1, 푝3 “ 푏, 푝4 “ ˆ휂1,´1, 푝5 “ ˆ휂0,2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' , 푝푁tot “ ˆ휂1,´푁2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The Fourier modes have been organized in a spiral fashion so that low frequency modes appear first in the list and ˆ휂1,0, ˆ휂0,1 have been replaced by 휏 and 푏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' see [73] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Our goal is to find 풑 given ˆ휂1,0 and ˆ휂0,1 such that Fr풑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ˆ휂1,0, ˆ휂0,1s “ 0, where we have re-ordered the arguments of F and R in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the computation, the function R is evaluated at 푀1 ˆ 푀2 grid points, hence there are 푀1푀2 equations, which are more than the number of unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For this reason, the nonlinear least-squares problem is overdetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The objective function F is computed from R by the trapezoidal rule approximation over T2, which is spectrally accurate, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11) 푓 p풑q “ 1 2푟p풑q푇푟p풑q « F r풑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ˆ휂1,0, ˆ휂0,1s , 푟푚p풑q “ R r풑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ˆ휂1,0, ˆ휂0,1s p훼푚1, 훼푚2q ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='푀1푀2 , ˜ 푚 “ 1 ` 푚1 ` 푀1푚2 훼푚푖 “ 2휋푚푖{푀푖 ¸ , 0 ď 푚푖 ă 푀푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The parameters 푝푗 are chosen to minimize 푓 p풑q using the Levenberg-Marquardt method [50, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The method requires a Jacobian matrix B푟푚{B푝푗, which we compute by solving the variational equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We have B푟푚 B푝푗 “ 훿Rp훼푚1, 훼푚2q{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='푀1푀2, where 푚 “ 1`푚1`푀1푚2 and the 푗th column of the Jacobian corresponds to setting 훿푝푗 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='10) to 1 and the others to 0 depending on the perturbation direction: 훿휏, 훿푏 or 훿 ˆ휂푗1,푗2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We compute quasi-periodic traveling solutions that bifurcate from the zero solution using 푁푥 “ 푁푦 “ 75 and 푀푥 “ 푀푦 “ 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We fix ˆ휂1,0 “ ˆ휂0,1 “ 10´5, choose ℎ to be the continuation parameter, and decrease ℎ from 3 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5 with Δℎ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='01 to obtain a family of quasi-periodic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In panel (a) of Figure 5, we plot the wave profile of the free surface for solutions at ℎ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5 and ℎ “ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The difference between these two solutions is small because they are both small-amplitude bifurcations from the zero solution for which we imposed the same amplitude parameters ˆ휂1,0 and ˆ휂0,1 at linear order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We stayed close to the linear regime in this example to investigate whether traveling solutions of the fully nonlinear equations, which we compute using the Levenberg-Marquardt method, behave as predicted by weakly nonlinear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' While the wave profiles are close to one another, the values of ℎ (3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5) and 휏 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='23088845108 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='0812490184995) differ substantially for the two solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In panel (b) of Figure 5, we plot the absolute value of the leading Fourier coefficients |ˆ휂2,0|, |ˆ휂0,2|, |ˆ휂1,1| and |ˆ휂1,´1| of the computed solutions as functions of ℎ, holding ˆ휂1,0 and ˆ휂0,1 fixed at 10´5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' These Fourier coefficients decrease as ℎ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In panel (c), we plot the absolute value of the divisors ˆ푆푗1,푗2 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='28) corresponding to these four Fourier coefficients, 24 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' WILKENING AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ZHAO Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Quasi-periodic traveling gravity-capillary waves bifurcating from the zero solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' (a) Surface elevation function of two solutions with ℎ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5 (dashed red line) and ℎ “ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='0 (solid black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' (b) Amplitudes of Fourier coefficients ˆ휂2,0, ˆ휂0,2, ˆ휂1,1, ˆ휂1,´1 of quasi-periodic traveling solutions for which ˆ휂1,0 and ˆ휂0,1 are fixed at 10´5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' (c) Absolute value of the corresponding divisors ˆ푆푗1,푗2 defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='28) in the weakly nonlinear model (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Here we solve (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11) by minimizing 푓 p풑q in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11) and check whether the solution behaves as predicted by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Panels (d) and (e) show |ˆ휂푗1,푗2| versus | ˆ푆푗1,푗2| for 2 ď |푗1| ` |푗2| ď 75 in the cases where ℎ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5 and ℎ “ 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' which decrease as ℎ decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The behavior of the Fourier coefficients and ˆ푆푗1,푗2 is consistent with the weakly nonlinear approximations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='32), where the Fourier coefficients are obtained through division by ˆ푆푗1,푗2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' As a result, smaller values of ˆ푆푗1,푗2 lead to larger Fourier coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Note that we are checking whether traveling solutions of the Euler equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11) obtained 10- 10° 20 10 25 10~6 10° 100 102 10410-10 10-15 10-20 10-25 10-2 100 102 104QUASI-PERIODIC WATER WAVES OF FINITE DEPTH 25 by minimizing 푓 p풑q « F r풑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ˆ휂1,0, ˆ휂0,1s in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='11) via the Levenberg-Marquardt method behave as predicted by the weakly nonlinear model (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='32);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' we did not solve (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='32) directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Panels (d) and (e) of Figure 5 demonstrate the relationship between |ˆ휂푗1,푗2| and | ˆ푆푗1,푗2| with 2 ď |푗1| ` |푗2| ď 75 for traveling solutions at ℎ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5 and ℎ “ 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Since the largest Fourier coefficients are fixed at 10´5, one expects roundoff errors around 10´20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' But instead the “roundoff floor,” visible in both panels, appears to grow linearly as | ˆ푆푗1,푗2| decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' This suggests that roundoff errors in the Levenberg-Marquardt method are amplified by the reciprocals of the divisors ˆ푆푗1,푗2 even though this is not a weakly nonlinear calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The “active” modes in which ˇˇˆ휂푗1,푗2 ˇˇ extends above the roundoff floor appear to be well-resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The plots look nearly identical if we refine the calculation, keeping 푁푥 “ 푁푦 “ 75 but increasing 푀푥 and 푀푦 from 200 to 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In fact, we plotted the data from this finer mesh in panels (d) and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In panel (e), when ℎ “ 3, there are just a few active modes ˆ휂푗1,푗2, and they all correspond to low frequency modes with 2 ď |푗1| ` |푗2| ď 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' But in panel (d), when ℎ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5, there are many active modes of both small and intermediate frequency, plotted with red and black markers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' This is consistent with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='38) and panel (c), where the small divisors from weakly nonlinear theory decrease as ℎ decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The fixed values ˆ휂1,0 “ ˆ휂0,1 “ 10´5 we selected for this calculation appear to be small enough when ℎ “ 3 that we could have computed the solution by weakly nonlinear theory, but large enough at ℎ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5 that it was necessary to solve the problem by the Levenberg-Marquardt approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Next we search for quasi-periodic bifurcations from finite-amplitude periodic traveling waves of finite depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We use a new procedure, described in detail for the case of deep water in [75], to locate bifurcation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Specifically, we use the signed smallest singular value 휒p푠q of the Jacobian J quap푠q, as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='18), as a bifurcation “test function” that changes sign at bifurcation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' When a zero of 휒p푠q is found, the kernel of the Jacobian J quap푠q of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='18) also furnishes a search direction 훿 ˜휂qua for the quasi-periodic branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We use ˜휂per ` 휖훿 ˜휂qua with an empirically chosen value of 휖 as the initial guess for the Levenberg-Marquardt solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We then use numerical continuation to follow this branch beyond the realm of linearization about the periodic traveling wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Instead of using ˆ휂1,0, ˆ휂0,1 and ℎ as continuation parameters, we use 휏, ℎ and the Fourier mode ˆ휂0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For simplicity, we hold 휏 and ℎ fixed and just vary the Fourier mode to obtain a one-parameter family of quasi-periodic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Figures 6 and 7 show two quasi-periodic gravity-capillary waves bifurcating from a branch of periodic traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The fluid depth in conformal space is ℎ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We set 휏 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='00327672209262 so that the first Fourier mode of the periodic waves resonates with the fifth Fourier mode, which corresponds to solutions of the Wilton ripple problem [1, 3, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For the periodic traveling wave, we set 푀 “ 300, 푁 “ 100 and use 푠 “ ˆ휂1 as the continuation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The 1D waves are computed on T and embedded in T2 when searching for bifurcations, so that ˆ휂1 becomes ˆ휂1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We computed periodic waves with amplitude 푠 ranging from 10´5 to 2 ˆ 10´4 with Δ푠 “ 10´5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' By tracking the sign of 휒p푠q, we find out that there is a zero of 휒p푠q when 푠 belongs to intervals r10´5, 2ˆ10´5s, r4ˆ10´5, 5ˆ10´5s, r7ˆ10´5, 8ˆ10´5s, r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1ˆ10´4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2ˆ10´4s and r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='7ˆ10´4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='8ˆ10´4s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We focus our discussion on the first and last intervals and locate the zeros of 휒p푠q in these intervals, which are the bifurcation points, using the numerical algorithm described in [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In double precision, the zeros and corresponding values of 휒 are (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='12) 푠1 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='83810709940 ˆ 10´5, 휒p푠1q “ ´7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='8 ˆ 10´15, 푠2 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='72625902886 ˆ 10´4, 휒p푠2q “ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='8 ˆ 10´15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 26 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' WILKENING AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ZHAO The periodic solutions at 푠1 and 푠2 are plotted with dotted black lines in panel (a) of Figures 6 and 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' These periodic solutions demonstrate the nonlinear interaction of Fourier modes of different wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Unlike the crests of sinusoidal waves, we observe small ripples at the wave peaks of the periodic wave at 푠1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' As the amplitude of the periodic solution increases, this nonlinear feature is more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For the periodic solution at 푠2, near 푥 “ 2휋푛 for 푛 P Z, there is a flat plateau with wave peaks shifted to the edges of the plateau, forming an interesting “cat ears” structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' These nonlinear features at the wave crests can be attributed to the effect of the capillary force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In panel (b) of Figures 6 and 7, we show contour plots of torus functions of these periodic traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We observe that the width of the yellow region is larger for the higher-amplitude periodic wave;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' in correspondence, this wave possesses wider wave crests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We compute secondary quasi-periodic bifurcation branches that intersect the primary pe- riodic branch at 푠1 and 푠2 and show the corresponding results in Figures 6 and 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In both computations, we set 푀푥 “ 300, 푀푦 “ 150, 푁푥 “ 100, 푁푦 “ 50 and use ˆ휂0,1 as the continuation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We follow the two quasi-periodic branches until ˆ휂0,1 “ 7 ˆ 10´5 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Quasi-periodic bifurcation from a periodic traveling gravity-capillary wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Panel (a) shows the periodic traveling wave where a bifurcation was found and the largest-amplitude solution we computed on the quasi-periodic bifurcation branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The dotted black line corresponds to the periodic wave and the red line corresponds to the quasi-periodic wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Panels (b) and (c) show contour plots of the torus functions of the periodic wave and the quasi-periodic wave, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The 1D quasi-periodic wave in panel (a) is extracted from the corresponding torus function along the characteristic lines of slope 푘 “ 1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 2, plotted with red dashed lines in panel (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=" T 2 0 0 2 4 T 0 T TTT 2 0 0 2 4 T 0 T T X10'QUASI-PERIODIC WATER WAVES OF FINITE DEPTH 27 Figure 7." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Quasi-periodic bifurcation from a larger-amplitude periodic traveling gravity-capillary wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The panels show the same information as in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' and ˆ휂0,1 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1 ˆ 10´4, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' the corresponding quasi-periodic traveling waves are plotted with red lines in panel (a) of Figures 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The objective function is minimized to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='14 ˆ 10´27 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='03 ˆ 10´28, respectively, for these solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In panel (a) of Figure 6, the os- cillations at the troughs of the quasi-periodic wave are ahead of the ones of the periodic wave near 휉 “ 3휋, 5휋, 7휋, 11휋 and are behind near 휉 “ 휋, which demonstrates the quasi-periodic feature of the secondary bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We also observe that the amplitude of the quasi-periodic wave in panel (a) of Figure 6 is noticeably larger than the periodic wave due to the activation of Fourier modes in the quasi-periodic direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In panel (c) of Figures 6 and 7, we show contour plots of the torus functions of the quasi-periodic traveling waves in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Unlike the periodic solution, the quasi-periodic solution depends on 훼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For example, one can see the variation of the yellow and blue regions in the 훼2 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Moreover, this variation is rather oscillatory in Figure 6, which adds to the difficulty of computing higher-amplitude quasi-periodic waves on the bifurcation branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' The 1D quasi-periodic waves are obtained by evaluating the corresponding torus functions along the the red dashed line of slope 1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In panel (a) of Figures 6 and 7, there will be crests if the dashed line in panel (c) passes through the yellow region and troughs if it passes through the blue region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Due to the variation in yellow region, the widths of the crests of the quasi-periodic wave are no longer constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For example, in panel (a) of Figure 7, the crests of the quasi-periodic wave are wider than those of the periodic wave near 휉 “ 6휋, 8휋 and narrower near 휉 “ 4휋, 10휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' T 2 0 2 0 4 9- 8 10 T 0 T T X10T 2 0 2 0 4 6 T 0 T T X1028 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' WILKENING AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' ZHAO 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Conclusion In this paper, we have presented a numerical study of two-dimensional finite-depth free surface waves in the spatially quasi-periodic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Specifically, we have studied both the initial value and traveling wave problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For the initial value problem, we derived the governing equations of water waves in the presence of a background flow and a non-flat bottom boundary in conformal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' As noted in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='7, the derivation is valid in both the quasi-periodic and periodic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Motivated by the experiments of Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' [62] studying spatially quasi-periodic surface waves in the presence of a quasi-periodic bottom boundary, we computed the time evolution of an initially flat surface with a background flow over a quasi-periodic bottom boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We also find that the waves develop quasi-periodic patterns in which the distance between adjacent wave peaks is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Next we computed spatially quasi-periodic traveling waves that bifurcate from the zero- amplitude wave or from finite-amplitude periodic traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Motivated by observations in [73, 75] that the Fourier coefficients of quasi-periodic traveling waves decay slower along certain directions, we derived the weakly nonlinear equations governing small-amplitude quasi-periodic traveling waves in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2 and found that there is a divisor ˆ푆푗1,푗2 in the formula for the Fourier coefficients ˆ휂푗1,푗2 of the weakly nonlinear solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' For example, in the case of deep water, this divisor reads (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1) ˆ푆푗1,푗2 “ 푔 푘 p|푗1 ` 푘푗2| ´ 푘qp|푗1 ` 푘푗2| ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Due to the unboundedness of 1{ ˆ푆푗1,푗2, the Fourier coefficients along directions |푗1 ` 푘푗2|´ 푘 “ 0 and |푗1 ` 푘푗2| ´ 1 “ 0 are expected to decay slower than in other directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We also study these divisors in the case of shallow water and find that weakly nonlinear theory breaks down faster when ℎ is smaller due to the factor of ℎ4 in the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='38) for ˆ푆푗1,푗2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the current work, we assume that the bottom boundary remains fixed in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the future, we plan to further extend our method to study quasi-periodic flows with a free surface over a moving bottom boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the case of periodic water waves, this has been studied in [57, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We also plan to analyze the linear stability of periodic traveling waves [19, 47, 52, 60] and investigate the long-time dynamics of traveling waves under unstable subharmonic perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In the quasi-periodic setting, we are able to compute the exact time evolution of these perturbed waves instead of their linearized approximations [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We are also interested in developing numerical methods, such as the Transformed Field Expansion method [48,49,54], to study the dynamics of these waves in three dimensions where the conformal mapping method no longer applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' On the theoretical side, a rigorous proof of the existence of quasi- periodic traveling waves is still an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We expect it will be necessary to employ a Nash-Moser iteration to tackle the small divisor problem, which has been successfully used to prove the existence of temporally quasi-periodic standing waves and traveling waves [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Funding: This work was supported in part by the National Science Foundation under award number DMS-1716560 and by the Department of Energy, Office of Science, Applied Scientific Computing Research, under award number DE-AC02-05CH11231;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' and by an NSERC (Canada) Discovery Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Declaration of interests: The authors report no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' QUASI-PERIODIC WATER WAVES OF FINITE DEPTH 29 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Computation of the conformal mapping from a infinite horizontal strip to the fluid domain In practice, the initial condition of the water wave is usually given in physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' There- fore, we need to compute the conformal mapping 푧p푤, 푡q to transform the initial condition from physical space to conformal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' As shown in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='23) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='24), the conformal mapping is determined by ℎ, 푥0, ˜휂s and ˜휂b, where 푥0 is fixed to be zero in the scope of this paper and ℎ, ˜휂s, ˜휂b are obtained by solving the following equations, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1) R1p훼1, 훼2q “ ˜휂s ´ ˜휂s,physp훼1 ` ˜휉s, 훼2 ` 푘 ˜휉sq “ 0, R2p훼1, 훼2q “ ˜휂b ´ ˜휂b,physp훼1 ` ˜휉b, 훼2 ` 푘 ˜휉bq “ 0, ˜휉s “ 퐻cothr˜휂ss ` 퐻cschr˜휂bs, ˜휉b “ ´퐻cschr˜휂ss ´ 퐻cothr˜휂bs, which come from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Moreover, we enforce the constraint ℎ “ ˆ휂s 0 ´ ˆ휂b 0 discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3 and rewrite ˜휂b as (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='2) ˜휂b “ ˆ휂s 0 ´ ℎ ` 푃r˜휂bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Otherwise problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='1) is underdetermined and the solution is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' In our computations, we consider ℎ and the Fourier coefficients of ˜휂s and ˜휂b as unknowns and define the following objective function (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3) Frℎ, ˆ휂s, ˆ휂bs : “ 1 8휋2 ż T2 R2 1rℎ, ˆ휂s, ˆ휂bs ` R2 2rℎ, ˆ휂s, ˆ휂bs 푑훼1 푑훼2 « 1 2푀1푀2 푀2´1 ÿ 푚2“0 푀1´1 ÿ 푚1“0 ” R2 1p2휋푚1{푀1, 2휋푚2{푀2q ` R2 1p2휋푚1{푀1, 2휋푚2{푀2q ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' We apply a Leveberg-Marquardt method [72] to solve the nonlinear least-squares prob- lem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='3) and compute the derivative of R1 and R2 with respect to the unknowns using the following variational equations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='4) 훿R1 “ 훿 ˜휂s ´ ˜휂s,phys 푥 훿 ˜휉s, 훿R2 “ 훿 ˆ휂s 0 ´ 훿ℎ ` 푃r훿 ˜휂bs ´ ˜휂b,phys 푥 훿 ˜휉b, 훿 ˜휉s “ 퐻cothr훿 ˜휂ss ` 퐻cschr훿 ˜휂bs ` ` 훿퐻coth˘ r˜휂ss ` ` 훿퐻csch˘ r˜휂bs, 훿 ˜휉b “ ´퐻cschr훿 ˜휂ss ´ 퐻cothr훿 ˜휂bs ´ ` 훿퐻csch˘ r˜휂ss ´ ` 훿퐻coth˘ r˜휂bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Here B푥 “ B푥1 ` 푘B푥2 and the symbols of 훿퐻coth and 훿퐻csch are (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content='5) 훿 ˆ퐻coth 푗1,푗2 “ 푖p푗1 ` 푘푗2q훿ℎ sinh2pp푗1 ` 푘푗2qℎq , 훿 ˆ퐻csch 푗1,푗2 “ ´푖p푗1 ` 푘푗2q cothpp푗1 ` 푘푗2qℎq cschpp푗1 ` 푘푗2qℎq훿ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' Akers and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfVvxj/content/2301.01289v1.pdf'} 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mode 100644 index 0000000000000000000000000000000000000000..e1ebef9400d9bedd6b2919d8b3ce5b00facf8d1f --- /dev/null +++ b/PdE2T4oBgHgl3EQfVgcw/content/tmp_files/2301.03823v1.pdf.txt @@ -0,0 +1,3097 @@ +arXiv:2301.03823v1 [math.NT] 10 Jan 2023 +On some analytic properties of a function associated with +the Selberg class satisfying certain special conditions +by +Hideto IWATA +Abstract. In 2001, M. Reko´s described the analytic behavior for a func- +tion f (z) connected with the Euler totient function for Im z > 0 (see (1.2)) +imitating the previous research of [1] and [3]. In the present paper, for +Im z > 0 we describe the analytic behavior of the generalized function +f (z, F) (see (2.1)), where the function F belongs to the subclass of the Sel- +berg class which has a polynomial Euler product and satisfies some special +conditions. +1. Introduction +J.Kaczorowski defined the associated Euler totient function for a class +of generalized L-functions including the Riemann zeta function, Dirichlet +L-functions and obtained an asymptotic formula (see [4]):By a polynomial +Euler product we mean a function F(s) of a complex variable s = σ + it +which for σ > 1 is defined by a product of the form +(1.1) +F(s) = +� +p +Fp(s) = +� +p +d +� +j=1 +� +1 − αj(p) +ps +�−1 +, +where p runs over primes and |αj(p)| ≤ 1 for all p and 1 ≤ j ≤ d. We +assume that d is chosen as small as possible, i.e. that there exists at least +one prime number p0 such that +d +� +j=1 +αj(p0) � 0. +Then d is called the Euler degree of F. Note that the L-functions from +number theory including the Riemann zeta function, Dirichlet L-functions, +Dedekind zeta and Hecke L-functions of algebraic number fields, as well as +the (normalized) L-functions of holomorphic modular forms and, conjec- +turally, many other L-functions are polynomial Euler products. +M. Reko´s described the analytic property of some function connected +with the Euler totient function (see [9]):We describe basic analytic proper- +ties of the function f (z) defined for Im z > 0 as follows : +(1.2) +f (z) = lim +n→∞ +� +ρ +0 0, meromorphic continuation to the whole complex plane, a certain +functional equation and information of singularities. The functional equa- +tion for f (z) connects the value of the function f at the points z and ¯z. Let +ℓ denote a smooth curve τ : [0, 1] −→ C such that τ(0) = −1 +4, τ(1) = 5 +2 and +0 < Im τ < 1 for t ∈ (0, 1). The analytic property of f (z) is described by the +following theorems: +Theorem 1.1 (Theorem 1. in [9]). The function f (z) is analytic on the upper +half-plane H and for z ∈ H we have +(1.3) +2πi f (z) = f1(z) + f2(z) − e +5 +2 z +∞ +� +n=1 +ϕ(n) +n +5 +2(z − log n) +where the last term on the right is a meromorphic function on the whole +complex plane with the poles at z = log n, n = 1, 2, . . .. The function +(1.4) +f1(z) = +� − 1 +4 +− 1 +4 +i∞ +ζ(s − 1) +ζ(s) +eszds +is analytic on H and +(1.5) +f2(z) = +� +ℓ(− 1 +4 , 5 +2 ) +ζ(s − 1) +ζ(s) +eszds +is analytic on the whole complex plane. +Theorem 1.2 (Theorem 2. in [9]). The function f (z) can be continued an- +alytically to a meromorphic function on the whole complex plane, which +satisfies the functional equation +(1.6) +f (z) + f (¯z) = B(z) +and +(1.7) +B(z) = − 6 +π2e2z+ 1 +2π2 +∞ +� +k,n=1 +µ(k) +n2k +� +1 +(nkez − 1)2 + +2 +nkez − 1 + +1 +(nkez + 1)2 − +2 +nkez + 1 +� +, +where B(z) is a meromorphic function on the whole complex plane with the +poles of the second order at z = − log nk, n, k = 1, 2, . . . and µ(n) is the +M¨obius function. +We see that the function f (z) has simple poles at z = log n (n = 1, 2, . . .) +with residue +−ϕ(n) +2πi , +where ϕ(n) is the Euler totient function. +Now we provide the definition of the Selberg class S as follows : F ∈ S +if +(i) (ordinary Dirichlet series) F(s) = +∞ +� +n=1 +aF(n)n−s, absolutely conver- +gent for σ > 1; +(ii) (analytic continuation) there exists an integer m ≥ 0 such that (s − +1)m · F(s) is an entire function of finite order; + +On some analytic. . . +3 +(iii) (functional equation) F(s) satisfies a functional equation of type +Φ(s) = ωΦ(1 − s), where +(1.8) +Φ(s) = Qs +r +� +j=1 +Γ(λ js + µj)F(s) = γ(s)F(s), +say, with r ≥ 0, Q > 0, λ j > 0, Re µj ≥ 0 and |ω| = 1; +(iv) (Ramanujan conjecture) for every ǫ > 0, aF(n) ≪ nǫ +(v) (Euler product) F(s) = +� +p +exp + +∞ +� +ℓ=0 +bF(pℓ) +pℓs +, where bF(n) = 0 un- +less n = pm with m ≥ 1, and bF(n) ≪ nϑ for some ϑ < 1 +2. +Note that we understand an empty product is equal to 1. +2. The extension of f (z) to the subclass of S +If a function F ∈ S has a polynomial Euler product (1.1), the subclass +of S of the functions with polynomial Euler product is denoted by Spoly. +Obtaining results similar to Theorem 1.4.1 and Theorem 1.4.2 for a func- +tion F belonging to Spoly is aim in this present paper. Now we assume +that (r, λ j) = (1, 1) for all j in the functional equation (1.8). The complex +number µ1 when r = 1 in (1.8) is hereafter referred to as µ. Let ρ denote +the non-trivial zeros of F with positive imaginary part. We assume that the +order of ρ is simple. Moreover, let Tn denote a sequence of real numbers +which yields appropriate grouping of the zeros which will be given later. +For Im z > 0, we consider a function defined by +(2.1) +f (z, F) = lim +n→∞ +� +ρ +0 0} and L denote the contour consisting of line +segments +[b, b + iTn] , [b + iTn, a + iTn] , [a + iTn, a] , +� +a, a + b +2 ++ iα +� +, +�a + b +2 ++ iα, b +� +, +where max +� +−3 +2, 1 +2 max {Re ρ; Re ρ < 0} +� +< a < 0, b > 5 +2. We assume that +the real part of s = a+it(t ∈ R) does not coincide the poles of Γ(s+µ)Γ(s−µ). +We consider the following contour integral round L : +(3.3) +� +L +ζ(s − 1) +F(s) eszds. +Since we assume the order of ρ is simple, we have by residue theorem +� +L +ζ(s − 1) +F(s) eszds = +� a +a+iTn +ζ(s − 1) +F(s) ezsds + +� +L +ζ(s − 1) +F(s) ezsds ++ +� b+iTn +b +ζ(s − 1) +F(s) ezsds + +� a+iTn +b+iTn +ζ(s − 1) +F(s) ezsds += 2πi +� +ρ +0 0), +������ +� b+iTn +a+iTn +ζ(s − 1) +F(s) ezsds +������ +≤ +� b+iTn +a+iTn +����� +ζ(s − 1) +F(s) ezs +����� |ds| +≪ +� b +a +|ζ(σ − 1 + iTn)| exp(C(log T)2 + xσ − yTn)dσ +≪ (b − a) exp{C(log T)2 − yTn + |x|(|a| + |b|)}T c +n, +(3.5) +where the constant c may depend on a, b. The last term on the right hand +side in the above tends to zero as n tends to infinity. By Theorem 4.1, +the convergence of the other integrals in (3.4) are ensured (see (4.5)-(4.7)). +Therefore, the series in (2.1) is convergent. □ + +On some analytic. . . +5 +4. Main theorems +In (3.4), we have as n tends to infinity +(4.1) +� a +a+i∞ +ζ(s − 1) +F(s) ezsds + +� +L +ζ(s − 1) +F(s) ezsds + +� b+i∞ +b +ζ(s − 1) +F(s) ezsds = 2πi f (z, F), +where the function f (z, F) is defined in (2.1). To calculate the integral along +the vertical line with s = b + it (t ≥ 0), we prepare the Dirichlet series +expansion of ζ(s − 1)/F(s) for σ > 2. +Definition 4.1 (p 34 in [4]). For σ > 1 and F ∈ Spoly, we define the function +µF as follows : +(4.2) +1 +F(s) = +∞ +� +n=1 +µF(n) +ns += +� +p +d +� +j=1 +� +1 − αj(p) +ps +� +. +Remark 4.2 (p34 in [4]). By (4.2), |µF(n)| ≤ τd(n), where τd(n) is the +divisor function of order d, so that ζd(s) = �∞ +n=1 τd(n)/ns for σ > 1. In +particular τ1(n) = 1 for all n. +Using (4.2), for σ > 2 +ζ(s − 1) +F(s) += + +∞ +� +l=1 +µF(l) +ls + + +∞ +� +m=1 +1 +ms−1 + += +∞ +� +n=1 +g(n) +ns , +(4.3) +where +(4.4) +g(n) = +� +d|n +µF(d)n +d. +Theorem 4.1. The function (2.1) is analytic on H and for z ∈ H we have +(4.5) +2πi f (z, F) = f1(z, F) + f2(z, F) − ebz +∞ +� +n=1 +g(n) +nb(z − log n), +where the last term on the right is a meromorphic function on the whole +complex plane with the poles at z = log n, n = 1, 2, . . .. The function +(4.6) +f1(z, F) = +� a +a+i∞ +ζ(s − 1) +F(s) eszds +is analytic on H and +(4.7) +f2(z, F) = +� +L +ζ(s − 1) +F(s) eszds +is analytic on the whole complex plane. +Theorem 4.2. For F belonging to Spoly whose (r, λ j) = (1, 1) for all j in +(1.8) and 0 ≤ µ < 1, the function (2.1) has a meromorphic continuation to +y > −π. + +6 +H. IWATA +The L-functions associated with holomorphic cusp forms and Dedekind +zeta functions of the imaginary quadratic fields are examples of F consid- +ering in Theorem 4.2. Let +(4.8) +H− = {z ∈ C : Im z < 0}. +We consider the function for z ∈ H− +(4.9) +f −(z, F) = lim +n→∞ +� +ρ +−Tn 0. □ +7. Proof of Theorem 4.2 +We prove that the function f (z, F)(z = x +iy) has a meromorphic contin- +uation to y > −π. By Theorem 4.1, the function +f1(z, F) = +� a +a+i∞ +ζ(s − 1) +F(s) ezsds += − +� a+i∞ +a +ζ(s − 1) +F(s) ezsds +is convergent for y > 0. We recall the hypotheses that (r, λ j) = (1, 1) for all +j in (1.8) and 0 ≤ µ < 1. We rewrite the functional equation (1.8) under +these hypotheses as follows : +QsΓ(s + µ)F(s) = ωQ1−sΓ(1 − s + µ)F(1 − s) += ωQ1−sΓ(1 − s + µ)F(1 − s), +where the conditions of Q and ω are the same as noted in (1.8). Hence +(7.1) +1 +F(s) = ωQ2s−1 +Γ(s + µ) +Γ(1 − s + µ) +1 +F(1 − s) +. +Using the following elementary formula for the Γ-function +Γ(s)Γ(1 − s) = +π +sin πs, + +On some analytic. . . +11 +(7.1) yields +(7.2) +1 +F(s) = ω +π Q2s−1 sin π(s − µ)Γ(s + µ)Γ(s − µ) +1 +F(1 − s) +. +By (7.2) and the functional equation for ζ(s), we have +f1(z, F) += − +� a+i∞ +a +ζ(s − 1) +F(s) ezsds += +ω +2π3Q +� a+i∞ +a +(2πQ2)s cos +� s +2π +� +Γ(2 − s)ζ(2 − s) sin π(s − µ) +× Γ(s + µ)Γ(s − µ) +ezs +F(1 − s) +ds += ωe−µπi +(2π)3Qi +� a+i∞ +a +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s − µ)Γ(s + µ)Γ(2 − s)e(z+ 3 +2 πi)sds +− +ωeµπi +(2π)3Qi +� a+i∞ +a +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s − µ)Γ(s + µ)Γ(2 − s)e(z− π +2 i)sds ++ ωe−µπi +(2π)3Qi +� a+i∞ +a +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s − µ)Γ(s + µ)Γ(2 − s)e(z+ π +2 i)sds +− +ωeµπi +(2π)3Qi +� a+i∞ +a +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s − µ)Γ(s + µ)Γ(2 − s)e(z− 3 +2 πi)sds += f11(z, F) + f12(z, F) + f13(z, F) + f14(z, F), +(7.3) +where f11(z, F), f12(z, F), f13(z, F), f14(z, F) denote the corresponding inte- +grals in (7.3) respectively. By Stirling’s formula +Γ(s + µ)Γ(s − µ)Γ(2 − s) ≍ e− 3 +2 π|t||t|a+ 1 +2 +as t tends to infinity and max +� +−3 +2, 1 +2 max{Re ρ; Re ρ < 0} +� +< a < 0, f11(z, F) +is analytic for y > −3π, f12(z, F) for y > −π, f13(z, F) for y > −2π, f14(z, F) +for y > 0. Splitting the integral in f14(z, F), we have +f14(z, F) = − ωeµπi +(2π)3Qi +� a+i∞ +a +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s − µ)Γ(s + µ)Γ(2 − s)e(z− 3 +2 πi)sds += ωeµπii +(2π)3Q +�� a+i∞ +a−i∞ +− +� a +a−i∞ +� +(2πQ2)s ζ(2 − s) +F(1 − s) +× Γ(s − µ)Γ(s + µ)Γ(2 − s)e(z− 3 +2 πi)sds. +We consider +(7.4) I1(z, F) = +� a+i∞ +a−i∞ +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s − µ)Γ(s + µ)Γ(2 − s)e(z− 3 +2 πi)sds +and +(7.5) I2(z, F) = +� a +a−i∞ +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s − µ)Γ(s + µ)Γ(2 − s)e(z− 3 +2 πi)sds. + +12 +H. IWATA +Here, we recall the hypothesis that the real part of s = a + it (t ∈ R) does +not coincide with the poles of Γ(s + µ)Γ(s − µ). By Stirling’s formula, +we can see that the integral I2(z, F) is convergent for y < 3π. Since the +function f14(z, F) was analytic for y > 0, the integral I1(z, F) is convergent +for 0 < y < 3π. Using the Dirichlet series expansion (4.2), we have formally +I1(z, F) = +� a+i∞ +a−i∞ +(2πQ2)s + +∞ +� +n=1 +1 +n2−s + + +∞ +� +k=1 +µF(k) +k1−s +Γ(s − µ)Γ(s + µ)Γ(2 − s)e(z− 3 +2 πi)sds += +∞ +� +k,n=1 +µF(k) +kn2 +� a+i∞ +a−i∞ +es{log(2πnkQ2)− 3 +2 πi+z}Γ(s − µ)Γ(s + µ)Γ(2 − s)ds. +(7.6) +The justification of the interchange of the order of integration and summa- +tion is ensured as follows : For 0 < y < 3π, we have +� a+i∞ +a−i∞ +����es{log(2nkπQ2)+z− 3 +2 πi}Γ(s − µ)Γ(s + µ)Γ(2 − s) +���� |ds| +≪ (nk)a · eax +� ∞ +−∞ +e−(y− 3 +2 π)t− 3 +2 π|t||t|a+ 1 +2dt += (nk)a · eax +�� 0 +−∞ +e−(y− 3 +2 π)t+ 3 +2πt(−t)a+ 1 +2dt + +� ∞ +0 +e−(y− 3 +2 π)t− 3 +2 πtta+ 1 +2dt +� +≪a,x,y (nk)a +≪a (nk)a. +Hence, we have +∞ +� +k,n=1 +������ +µF(k) +kn2 +� a+i∞ +a−i∞ +es{log(2nkπQ2)+z− 3 +2 πi}Γ(s − µ)Γ(s + µ)Γ(2 − s)ds +������ ≪a +∞ +� +k,n=1 +������ +µF(k) +k1−an2−a +������ . +Since the series for k is absolutely convergent by (4.2), the above series of +the left hand side is convergent absolutely and uniformly. Therefore, the +interchange of the order of integration and summation is justified for 0 < +y < 3π. Let m1, m2 be non-negative integers. The residue of the integrand +in (7.6) at s = µ − m1 is +R(1) +k,n,m1(z, µ) = +lim +s→µ−m1{s − (µ − m1)}Γ(s − µ)Γ(s + µ)Γ(2 − s)e{log(2πnkQ2)− 3 +2 πi+z}s += (−1)m1 +m1! Γ(2µ − m1)Γ(2 − µ + m1)(2πnkQ2)µ−m1e(z− 3 +2 πi)(µ−m1). +(7.7) +Similarly, the residue of the integrand in (7.6) at s = −µ − m2 is +(7.8) +R(1) +k,n,m2(z, µ) = (−1)m2 +m2! Γ(−2µ − m2)Γ(2 + µ + m2)(2πnkQ2)−µ−m2e(z− 3 +2 πi)(−µ−m2). +When µ = 0, {Γ(s)}2 has a double pole at s = −m, where m is a non-negative +integer. For every positive ǫ, using the Taylor expansion for the every factor +of the integrand in (7.6) at s = −m + ǫ, the residue of the integrand in (7.6) + +On some analytic. . . +13 +at s = −m is +(7.9) +R(1) +k,n,m,0(z) = m + 1 +m! +e−(z− 3 +2 πi)m +(2πnkQ2)m +log(2πnkQ2) + z − 3 +2πi + +m +� +k1=1 +1 +k1 +− C0 − +1 +m + 1 + . +Similarly, when µ = 1 +2, Γ +� +s − 1 +2 +� +Γ +� +s + 1 +2 +� += +� +s − 1 +2 +� � +Γ +� +s − 1 +2 +��2 has a dou- +ble pole at s = 1 +2 − m. In the same way, the residue of the integrand in (7.6) +at s = 1 +2 − m is +R(1) +k,n,m, 1 +2(z) = +Γ +� 3 +2 + m +� +(m!)2 +e( 1 +2−m)(z− 3 +2 πi) +(2πnkQ2)m− 1 +2 +� +m +� +ψ +�3 +2 + m +� +− 2ψ(m + 1) +� +− m +� +log(2πnkQ2) + z − 3 +2πi +� ++ 1 +� +, +(7.10) +where ψ(s) is the logarithmic derivative of Γ(s), i.e. +(7.11) +ψ(s) := Γ′ +Γ (s). +Since for 0 < y < 3π integrals along the upper and the lower side of the +contour tend to 0, for µ except µ � 0, 1 +2, we have by theorem of residues +� a+i∞ +a−i∞ +es{log(2nkπQ2)+z− 3 +2 πi}Γ(s − µ)Γ(s + µ)Γ(2 − s)ds += − +� +C +es{log(2nkπQ2)+z− 3 +2 πi}Γ(s − µ)Γ(s + µ)Γ(2 − s)ds +− 2πi + +M +� +m1=0 +R(1) +k,n,m1(z, µ) + +M′ +� +m2=0 +R(1) +k,n,m2(z, µ) + , +(7.12) +where C is the contour which the poles of Γ(2−s) and those of Γ(s+µ)Γ(s− +µ) are on opposite sides of it. Of course, when µ = 0, 1 +2, the terms of residue +in (7.12) are replaced by +(7.13) +M′′ +� +m=0 +R(1) +k,n,m,0(z), +M′′ +� +m=0 +R(1) +k,n,m, 1 +2(z). +We use the same convention hereafter. Putting w = 2 − s in the integral +round the contour C on the right hand side in (7.12) and using Barnes type +integral for the Whittaker function (5.6), for µ which is not integer except +zero and |y − 3 +2π| < 3 +2π, the integral round the contour C in (7.12) is +(7.14) +2πi(2πnkQ2) +1 +2 exp + +e +3 +2 πi−z +4πnkQ2 + z +2 − 3 +4πi + Γ(2 − µ)Γ(2 + µ)W− 3 +2 ,µ + +e +3 +2 πi−z +2πnkQ2 + . +Therefore, we have +I1(z, F) = +∞ +� +k,n=1 +µF(k) +kn2 +2πi · (2πnkQ2) +1 +2 exp + +e +3 +2 πi−z +4πnkQ2 + z +2 − 3 +4πi + +× Γ(2 − µ)Γ(2 + µ)W− 3 +2 ,µ + +e +3 +2 πi−z +2πnkQ2 + − 2πi + +M +� +m1=0 +R(1) +k,n,m1(z, µ) + +M′ +� +m2=0 +R(1) +k,n,m2(z, µ) + + . +(7.15) +The following lemma ensures the convergence for the series on the right +hand side in (7.15). + +14 +H. IWATA +Lemma 7.1. The series on the right hand side in (7.15) is absolutely and +uniformly convergent on every compact subset on the whole complex plane. +We will prove Lemma 7.1 in the next section. By Lemma 7.1, for F ∈ +Spoly whose (r, λ j) = (1, 1) for all j in (1.8) and 0 ≤ µ < 1, we have the +following analytic continuation of f1(z, F) for y > −π : +f1(z, F) = ωe−µπi +(2π)3Qi +� a+i∞ +a +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s − µ)Γ(s + µ)Γ(2 − s)e(z+ 3 +2 πi)sds +− +ωeµπi +(2π)3Qi +� a+i∞ +a +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s − µ)Γ(s + µ)Γ(2 − s)e(z− π +2 i)sds ++ ωe−µπi +(2π)3Qi +� a+i∞ +a +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s − µ)Γ(s + µ)Γ(2 − s)e(z+ π +2 i)sds ++ ωeµπii +(2π)3Q +∞ +� +k,n=1 +µF(k) +kn2 +× 2πi +(2πnkQ2) +1 +2 exp + +e +3 +2 πi−z +4πnkQ2 + z +2 − 3 +4πi + +× Γ(2 − µ)Γ(2 + µ)W− 3 +2 ,µ + +e +3 +2 πi−z +2πnkQ2 + − +M +� +m1=0 +R(1) +k,n,m1(z, µ) − +M′ +� +m2=0 +R(1) +k,n,m2(z, µ) + ++ +ωeµπi +(2π)3Qi +� a +a−i∞ +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s − µ)Γ(s + µ)Γ(2 − s)e(z− 3 +2 πi)sds. +(7.16) +The first is analytic for y > −3π, the second for y > −π, the third for +y > −2π, the fourth is analytic on the whole complex plane by Lemma 7.1, +and the next is analytic for y < 3π. Therefore, (7.16) completes the proof of +the continuation of f (z, F) to the region y > −π. □ +8. Proof of Lemma 7.1 +We prove Lemma 7.1. First, we consider the case µ > 1 +2. By the asymp- +totic expansion (5.12), +W− 3 +2 ,µ + +e +3 +2πi−z +2πnkQ2 + = +Γ(2µ) +Γ(2 + µ) + +e +3 +2πi−z +2πnkQ2 + +1 +2 −µ ++ O + +|e( 3 +2 πi−z)( 3 +2 −µ)| +(2πnkQ2) +3 +2−µ + . +Hence, the inside of the curly brackets on the right hand side of (7.15) is +2πi(2πnkQ2) +1 +2 exp + +e +3 +2 πi−z +4πnkQ2 + z +2 − 3 +4πi + Γ(2 − µ)Γ(2 + µ)W− 3 +2 ,µ + +e +3 +2 πi−z +2πnkQ2 + +−2πi + +M +� +m1=0 +R(1) +k,n,m1(z, µ) + +M′ +� +m2=0 +R(1) +k,n,m2(z, µ) + += 2πi(2πnkQ2)µ exp + +e +3 +2 πi−z +4πnkQ2 − 3 +2µπi + µz + Γ(2 − µ)Γ(2µ) ++OQ,µ,x +� +1 +(2πnkQ2)1−µ exp +� +e−x +4πnkQ2 +�� +−2πi + +M +� +m1=0 +R(1) +k,n,m1(z, µ) + +M′ +� +m2=0 +R(1) +k,n,m2(z, µ) + . +Now, by the Taylor expansion +exp + +e +3 +2 πi−z +4πnkQ2 + = 1 + O + +������ +e +3 +2 πi−z +4πnkQ2 +������ + + +On some analytic. . . +15 += 1 + O +� +e−x +4πnkQ2 +� +(8.1) +as n, k tend to infinity and (7.7), (7.8), we have +2πi(2πnkQ2) +1 +2 exp + +e +3 +2 πi−z +4πnkQ2 + z +2 − 3 +4πi + Γ(2 − µ)Γ(2 + µ)W− 3 +2 ,µ + +e +3 +2 πi−z +2πnkQ2 + +−2πi + +M +� +m1=0 +R(1) +k,n,m1(z, µ) + +M′ +� +m2=0 +R(1) +k,n,m2(z, µ) + += OQ,µ,x +� +1 +(nk)1−µ +� ++ OQ,µ,x +� +1 +(nk)1−µ + +1 +(nk)2−µ +� +−2πi + +M +� +m1=1 +R(1) +k,n,m1(z, µ) + +M′ +� +m2=0 +R(1) +k,n,m2(z, µ) + +≪ OQ,µ,x +� +1 +(nk)1−µ +� ++ + +M +� +m1=1 +���R(1) +k,n,m1(z, µ) +��� + +M′ +� +m2=0 +���R(1) +k,n,m2(z, µ) +��� + +≪Q,µ,x OQ,µ,x +� +1 +(nk)1−µ +� ++ +M +� +m1=1 +1 +(nk)m1−µ + +M′ +� +m2=0 +1 +(nk)m2+µ += OQ,µ,x +� +1 +(nk)1−µ +� +Hence, I1(z, F) is evaluated as follows : +I1(z, F) ≪Q,µ,x +∞ +� +k,n=1 +|µF(k)| +kn2 +· OQ,µ,x +� +1 +(nk)1−µ +� += OQ,µ,x + +∞ +� +k,n=1 +|µF(k)| +k2−µn3−µ + . +Therefore, the series on the right hand side in (7.15) is convergent for 1 +2 < +µ < 1. +Secondly, in the case µ = 1 +2, by (5.13), +W− 3 +2 , 1 +2 + +e +3 +2 πi−z +2πnkQ2 + = +1 +Γ +�5 +2 +� + Oy +� +e−x +2πnkQ2 log +� +e−x +2πnkQ2 +�� += +1 +Γ +�5 +2 +� + Oy + +� +e−x +2πnkQ2 +�1−δ , +where δ is any positive real number. By the same calculation as in the first +case and using (7.10), the inside of the curly brackets on the right hand side +of (7.15) is evaluated as follows : +2πi(2πnkQ2) +1 +2 exp + +e +3 +2 πi−z +4πnkQ2 + z +2 − 3 +4πi + Γ +�3 +2 +� +Γ +�5 +2 +� +W− 3 +2 , 1 +2 + +e +3 +2 πi−z +2πnkQ2 + +−2πi +M′′ +� +m=0 +R(1) +k,n,m, 1 +2(z) + +16 +H. IWATA +≪ OQ,x,y +� +1 +(nk) +1 +2−δ +� ++ +M′′ +� +m=1 +log nk +(nk)m− 1 +2 += OQ,x,y,M′′ +� +1 +(nk) +1 +2 −δ +� +. +Hence, I1(z, F) is evaluated as follows : +I1(z, F) ≪ +∞ +� +k,n=1 +|µF(k)| +kn2 +· OQ,x,y,M′′ +� +1 +(nk) +1 +2 −δ +� += OQ,x,y,M′′ + +∞ +� +k,n=1 +|µF(k)| +k +3 +2 −δn +5 +2 −δ + . +Therefore, the series on the right hand side in (7.15) is convergent for µ = 1 +2. +Thirdly, in the case 0 < µ < 1 +2, by (5.14), +W− 3 +2,µ + +e +3 +2 πi−z +2πnkQ2 + = +Γ(2µ) +Γ(2 + µ) + +e +3 +2 πi−z +2πnkQ2 + +1 +2−µ ++ O + +e−x(µ+ 1 +2) +(2πnkQ2)µ+ 1 +2 + +and +2πi(2πnkQ2) +1 +2 exp + +e +3 +2 πi−z +4πnkQ2 + z +2 − 3 +4πi + Γ(2 − µ)Γ(2 + µ)W− 3 +2 ,µ + +e +3 +2 πi−z +2πnkQ2 + +−2πi + +M +� +m1=0 +R(1) +k,n,m1(z, µ) + +M′ +� +m2=0 +R(1) +k,n,m2(z, µ) + +≪Q,µ,x OQ,µ,x +� +1 +(nk)1−µ +� ++ +M +� +m1=1 +1 +(nk)m1−µ + +M′ +� +m2=0 +1 +(nk)m2+µ += OQ,µ,x +� +1 +(nk)1−µ +� +. +Therefore, the series on the right hand side in (7.15) is convergent for 0 < +µ < 1 +2. +Finally, in the case µ = 0, by (5.16), +W− 3 +2 ,0 + +e +3 +2 πi−z +2πnkQ2 + = − +e +3 +4 πi− z +2 +(2πnkQ2) +1 +2 +� +log +� +e−x +2πnkQ2 +� ++ +�3 +2π − y +� +i + Γ′ +Γ (2) + 2C0 +� ++ OQ,x +� +1 +(nk) +3 +2 log +� +e−x +2πnkQ2 +�� +. +Using the recurrence formula +(8.2) +ψ(s + 1) = 1 +s + ψ(s) +(see [6]) and (7.9), the inside of the curly brackets on the right hand side of +(7.15) is evaluated as follows : +2πi(2πnkQ2) +1 +2 exp + +e +3 +2 πi−z +4πnkQ2 + z +2 − 3 +4πi + Γ(2)2W− 3 +2 ,0 + +e +3 +2 πi−z +2πnkQ2 + − 2πi +M′′ +� +m=0 +R(1) +k,n,m,0(z) + +On some analytic. . . +17 += −2πi +�Γ′ +Γ (2) + C0 − 1 +� ++ OQ,x,y +� 1 +nk log +� +e−x +2πnkQ2 +�� +− 2πi +M′′ +� +m=1 +R(1) +k,n,m,0(z) += OQ,x,y +� 1 +nk log +� +e−x +2πnkQ2 +�� +− 2πi +M′′ +� +m=1 +R(1) +k,n,m,0(z) += OQ,x,y +� 1 +nk · +1 +(nk)1−δ +� +− 2πi +M′′ +� +m=1 +R(1) +k,n,m,0(z) += OQ,x,y +� +1 +(nk)2−δ +� +− 2πi +M′′ +� +m=1 +R(1) +k,n,m,0(z) +≪Q,x,y OQ,x,y +� +1 +(nk)2−δ +� ++ +M′′ +� +m=1 +log nk +(nk)m +≪ OM′′ +� +1 +(nk)1−δ +� +, +where δ is any positive real number. Hence, I1(z, F) is evaluated as follows +: +I1(z, F) ≪M′′ +∞ +� +k,n=1 +|µF(k)| +kn2 +· +1 +(nk)1−δ += +∞ +� +k,n=1 +|µF(k)| +k2−δn3−δ. +Therefore, the series on the right hand side in (7.15) is convergent for µ = 0. +In summary, +I1(z, F) ≤ +∞ +� +k,n=1 +|µF(k)| +kn2 +· max +� +1 +(nk)1−µ, +1 +(nk) +1 +2 −δ +� +. +(8.3) +By (5.2),(5.3) and (5.4), the Whittaker function W− 3 +2 ,µ +� +e +3 +2 πi−z +2πnkQ2 +� +is analytic for +all z ∈ C. Therefore, We have the desired result. □ +9. Proof of Corollary 4.3 and another proof +We prove Corollary 4.3. We use the following lemma similar to Lemma +3.1. We can prove this lemma by modifying the proof of Lemma 3.1. +Lemma 9.1. Let F ∈ S and let T be sufficiently large. Moreover, let H = +D log log T be fixed, where D is a large positive constant. In any subinterval +of length 1 in [−T − H, −T + H] there are lines t = t0 such that +(9.1) +|F(σ + it0)|−1 = O(exp(C(log T)2)) +uniformly in σ ≥ −2. +We consider the integral +(9.2) +� +L ′ +ζ(s − 1) +F(s) ezsds, +where L ′ is the contour symmetrical upon the real axis to L in (3.3). By +Lemma 9.1, the integral along the lower side of the contour tends to 0 as + +18 +H. IWATA +n tends to infinity for z ∈ H−. Then, we have by residue theorem and the +definition (4.9), in a similar manner as (4.1), +(9.3) +2πi f −(z, F) = f − +1 (z, F) + f − +2 (z, F) + f − +3 (z, F), +where +(9.4) +f − +1 (z, F) = +� a−i∞ +a +ζ(s − 1) +F(s) eszds +is analytic on H−, +(9.5) +f − +2 (z, F) = +� +L +ζ(s − 1) +F(s) eszds +is analytic on the whole complex plane. We consider the same setting as in +L for the curve L. In the same way as obtaining (6.1), +f − +3 (z, F) = +� b +b−i∞ + +∞ +� +n=1 +g(n) +ns + ezsds += +∞ +� +n=1 +g(n) +� b +b−i∞ +es(z−log n)ds += ebz +∞ +� +n=1 +g(n) +nb(z − log n) +(9.6) +is meromorphic on the whole complex plane. Now we already know that +f − +1 (z, F) is analytic for y < 0, and we have to continue to y < π just as in +the case of f1(z, F) (see Section 7). By the functional equation for ζ(s) and +F(s), we have +f − +1 (z, F) = +� a−i∞ +a +ζ(s − 1) +F(s) ezsds += − +� a +a−i∞ +ζ(s − 1) +F(s) ezsds += f − +11(z, F) + f − +12(z, F) + f − +13(z, F) + f − +14(z, F), +(9.7) +where +(9.8) +f − +11(z, F) = ωe−µπi +(2π)3Qi +� a +a−i∞ +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s+µ)Γ(s−µ)Γ(2−s)e(z+ 3 +2 πi)sds +is analytic for y < 0, +(9.9) +f − +12(z, F) = − ωeµπi +(2π)3Qi +� a +a−i∞ +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s+µ)Γ(s−µ)Γ(2−s)e(z− π +2 i)sds +for y < 2π, +(9.10) +f − +13(z, F) = ωe−µπi +(2π)3Qi +� a +a−i∞ +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s +µ)Γ(s −µ)Γ(2 − s)e(z+ π +2 i)sds + +On some analytic. . . +19 +for y < π, +(9.11) +f − +14(z, F) = − ωeµπi +(2π)3Qi +� a +a−i∞ +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s+µ)Γ(s−µ)Γ(2−s)e(z− 3 +2 πi)sds +for y < 3π. Splitting the integral on the right hand side in (8.8) just as in the +case of f14(z, F), we have +f − +11(z, F) = I− +1 (z, F) + I− +2 (z, F), +where +(9.12) +I− +1 (z, F) = ωe−µπi +(2π)3Qi +� a+i∞ +a−i∞ +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s+µ)Γ(s−µ)Γ(2−s)e(z+ 3 +2 πi)sds +and +(9.13) +I− +2 (z, F) = − ωe−µπi +(2π)3Qi +� a+i∞ +a +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s+µ)Γ(s−µ)Γ(2−s)e(z+ 3 +2 πi)sds. +We see that the integral I− +2 (z, F) is convergent for y > −3π by the same +way as in (7.5). Since f − +11(z, F) is analytic for y < 0, the integral I− +1 (z, F) is +convergent for −3π < y < 0 and we can calculate I− +1 (z, F) for −3π < y < 0 +in a similar way as (7.4) (Section 7). Let m1, m2 and m be non-negative +integers. By taking the path of integration C in (7.12), we have for µ which +is not integer except zero and |y + 3 +2π| < 3 +2π +I− +1 (z, F) = ωe−µπi +(2π)3Qi +∞ +� +k,n=1 +µF(k) +kn2 +× 2πi +(2πnkQ2) +1 +2 exp + +3 +4πi + z +2 + e− 3 +2 πi−z +4πnkQ2 + +× Γ(2 + µ)Γ(2 − µ)W− 3 +2 , µ + +e− 3 +2 πi−z +2πnkQ2 + − +M +� +m1=0 +R(2) +k,n,m1(z, µ) − +M′ +� +m2=0 +R(2) +k,n,m2(z, µ) +, +(9.14) +where +R(2) +k,n,m1(z, µ) = (−1)m1 +m1! Γ(2µ − m1)Γ(2 − µ + m1)(2πnkQ2)µ−m1e(z+ 3 +2 πi)(µ−m1), +(9.15) +R(2) +k,n,m2(z, µ) = (−1)m2 +m2! Γ(−2µ − m2)Γ(2 + µ + m2)(2πnkQ2)−µ−m2e(z+ 3 +2 πi)(−µ−m2), +(9.16) +R(2) +k,n,m,0(z) = m + 1 +m! +e−(z+ 3 +2 πi)m +(2πnkQ2)m +log(2πnkQ2) + z + 3 +2πi + +m +� +k1=1 +1 +k1 +− C0 − +1 +m + 1 + +(9.17) +and +R(2) +k,n,m, 1 +2(z) = +Γ +� 3 +2 + m +� +(m!)2 +e( 1 +2−m)(z+ 3 +2 πi) +(2πnkQ2)m− 1 +2 +� +m +� +ψ +�3 +2 + m +� +− 2ψ(m + 1) +� +− m +� +log(2πnkQ2) + z + 3 +2πi +� ++ 1 +� +(9.18) +are residues of the integrand in (9.12) at s = µ − m1, −µ − m2, −m and +1 +2 − m respectively. The convergence of the series on the right hand side in +(9.14) follows in a similar manner as the consideration in (7.15). Finally, + +20 +H. IWATA +by (9.7)-(9.18) we obtain the following continuation of f − +1 (z, F) to y < π : +For F ∈ Spoly whose (r, λ j) = (1, 1) for all j in (1.8) and 0 ≤ µ < 1, +f − +1 (z, F) = ωe−µπi +(2π)3Qi +∞ +� +k,n=1 +µF(k) +kn2 +× 2πi +(2πnkQ2) +1 +2 exp + +3 +4πi + z +2 + e− 3 +2 πi−z +4πnkQ2 + +× Γ(2 + µ)Γ(2 − µ)W− 3 +2 , µ + +e− 3 +2 πi−z +2πnkQ2 + − +M +� +m1=0 +R(2) +k,n,m1(z, µ) − +M′ +� +m2=0 +R(2) +k,n,m2(z, µ) + +− ωe−µπi +(2π)3Qi +� a+i∞ +a +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s + µ)Γ(s − µ)Γ(2 − s)e(z+ 3 +2 πi)sds +− +ωeµπi +(2π)3Qi +� a +a−i∞ +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s + µ)Γ(s − µ)Γ(2 − s)e(z− π +2 i)sds ++ ωe−µπi +(2π)3Qi +� a +a−i∞ +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s + µ)Γ(s − µ)Γ(2 − s)e(z+ π +2 i)sds +− +ωeµπi +(2π)3Qi +� a +a−i∞ +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s + µ)Γ(s − µ)Γ(2 − s)e(z− 3 +2 πi)sds. +(9.19) +Since a lemma similar to Lemma7.1 holds for the series on the right hand +side in (9.19), the series we now consider is also absolutely and uniformly +convergent on every compact subset on the whole complex plane. There- +fore, we complete the continuation of f −(z, F) analytic for y < 0 to the +region y < π. +Also, Corollary4.3 can be proved form Theorem4.2 and the definition +(4.9) directly as follows : For z ∈ H− and ρ with Im ρ < 0, +f −(z, F) = +� +ρ +eρzζ(ρ − 1) +F′(ρ) += +� +ρ +ζ(ρ − 1) +F′(ρ) +eρz += +� +ρ′ +ζ(ρ′ − 1) +F′ � +ρ′� eρ′z, +where ρ′ = ρ. We recall the definition F(s) = F(s) for F ∈ S. Hence, the +sum on the right hand side in the third line yields +� +ρ′ +ζ(ρ′ − 1) +F′(ρ′) +eρ′z = f (z, F). +Here, we use the fact that if F ∈ Spoly, then so is F ∈ Spoly. Of course, +Spoly may be replaced by S. By Theorem 4.2, the function f (z, F) has a +meromorphic continuation to y > −π. Hence f (z, F) has a meromorphic +continuation to y < π. +10. Proof of Theorem 4.4 +We add (7.16) to (9.19). Since some integrals are canceled, we have for +|y| < π +f1(z, F) + f − +1 (z, F) = f11(z, F) + f12(z, F) + f13(z, F) + +ωeµπi +(2π)3QiI1(z, F) + +ωeµπi +(2π)3QiI2(z, F) ++ I− +1 (z, F) + I− +2 (z, F) + f − +12(z, F) + f − +13(z, F) + f − +14(z, F) + +On some analytic. . . +21 += ωeµπii +(2π)3Q +∞ +� +k,n=1 +µF(k) +kn2 +× 2πi +(2πnkQ2) +1 +2 exp +−3 +4πi + z +2 + e +3 +2 πi−z +4πnkQ2 + +× Γ(2 + µ)Γ(2 − µ)W− 3 +2 , µ + +e +3 +2 πi−z +2πnkQ2 + − +M +� +m1=0 +R(1) +k,n,m1(z, µ) − +M′ +� +m2=0 +R(1) +k,n,m2(z, µ) + ++ ωe−µπi +(2π)3Qi +∞ +� +k,n=1 +µF(k) +kn2 +× 2πi +(2πnkQ2) +1 +2 exp + +3 +4πi + z +2 + e− 3 +2 πi−z +4πnkQ2 + +× Γ(2 + µ)Γ(2 − µ)W− 3 +2 , µ + +e− 3 +2 πi−z +2πnkQ2 + − +M +� +m1=0 +R(2) +k,n,m1(z, µ) − +M′ +� +m2=0 +R(2) +k,n,m2(z, µ) + ++ A1(z, F) + A2(z, F), +where +(10.1) +A1(z, F) = − ωeµπi +(2π)3Qi +� a+i∞ +a−i∞ +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s+µ)Γ(s−µ)Γ(2−s)e(z− π +2 i)sds +and +(10.2) +A2(z, F) = ωe−µπi +(2π)3Qi +� a+i∞ +a−i∞ +(2πQ2)s ζ(2 − s) +F(1 − s) +Γ(s+µ)Γ(s−µ)Γ(2−s)e(z+ π +2 i)sds. +The integrals A1(z, F) and A2(z, F) are convergent for |y| < π and we can +obtain series expressions of them involving Whittaker functions in a way +similar to the case of I1(z, F) and I− +1 (z, F). We have for |y| < π, F ∈ Spoly +whose (r, λ j) = (1, 1) for all j in (1.8) and 0 ≤ µ < 1 +A1(z, F) = − ωeµπi +(2π)3Qi +∞ +� +k,n=1 +µF(k) +kn2 × 2πi +� +(2πnkQ2) +1 +2 exp +� +−π +4i + z +2 + +e +π +2 i−z +4πnkQ2 +� +× Γ(2 + µ)Γ(2 − µ)W− 3 +2, µ +� e +π +2 i−z +2πnkQ2 +� +− +M +� +m1=0 +R(3) +k,n,m1(z, µ) − +M′ +� +m2=0 +R(3) +k,n,m2(z, µ) + , +where +R(3) +k,n,m1(z, µ) = (−1)m1 +m1! Γ(2µ − m1)Γ(2 − µ + m1)(2πnkQ2)µ−m1e(z− π +2 i)(µ−m1), +(10.3) +R(3) +k,n,m2(z, µ) = (−1)m2 +m2! Γ(−2µ − m2)Γ(2 + µ + m2)(2πnkQ2)−µ−m2e(z− π +2 i)(−µ−m2), +(10.4) +R(3) +k,n,m,0(z) = m + 1 +m! +e−(z− π +2 i)m +(2πnkQ2)m +log(2πnkQ2) + z − π +2i + +m +� +k1=1 +1 +k1 +− C0 − +1 +m + 1 + +(10.5) +and +R(3) +k,n,m, 1 +2(z) = +Γ +� 3 +2 + m +� +(m!)2 +e( 1 +2 −m)(z− π +2 i) +(2πnkQ2)m− 1 +2 +� +m +� +ψ +�3 +2 + m +� +− 2ψ(m + 1) +� +− m +� +log(2πnkQ2) + z − π +2i +� ++ 1 +� +(10.6) + +22 +H. IWATA +are residues of the integrand in A1(z, F) at s = µ−m1, −µ−m2, −m and 1 +2 −m +respectively. Similarly, for |y| < π, F ∈ Spoly whose (r, λ j) = (1, 1) for all j +in (1.8) and 0 ≤ µ < 1 +A2(z, F) = ωe−µπi +(2π)3Qi +∞ +� +k,n=1 +µF(k) +kn2 × 2πi +� +(2πnkQ2) +1 +2 exp +�π +4i + z +2 + e− π +2 i−z +4πnkQ2 +� +× Γ(2 + µ)Γ(2 − µ)W− 3 +2, µ +� e− π +2 i−z +2πnkQ2 +� +− +M +� +m1=0 +R(4) +k,n,m1(z, µ) − +M′ +� +m2=0 +R(4) +k,n,m2(z, µ) + , +where +R(4) +k,n,m1(z, µ) = (−1)m1 +m1! Γ(2µ − m1)Γ(2 − µ + m1)(2πnkQ2)µ−m1e(z+ π +2 i)(µ−m1), +(10.7) +R(4) +k,n,m2(z, µ) = (−1)m2 +m2! Γ(−2µ − m2)Γ(2 + µ + m2)(2πnkQ2)−µ−m2e(z+ π +2 i)(−µ−m2), +(10.8) +R(4) +k,n,m,0(z) = m + 1 +m! +e−(z+ π +2 i)m +(2πnkQ2)m +log(2πnkQ2) + z + π +2i + +m +� +k1=1 +1 +k1 +− C0 − +1 +m + 1 + +(10.9) +and +R(4) +k,n,m, 1 +2(z) = +Γ +� 3 +2 + m +� +(m!)2 +e( 1 +2 −m)(z+ π +2 i) +(2πnkQ2)m− 1 +2 +� +m +� +ψ +�3 +2 + m +� +− 2ψ(m + 1) +� +− m +� +log(2πnkQ2) + z + π +2i +� ++ 1 +� +(10.10) +are residues of the integrand in A2(z, F) at s = µ−m1, −µ−m2, −m and 1 +2 −m +respectively. Finally, for |y| < π, F ∈ Spoly whose (r, λ j) = (1, 1) for all j in +(1.8) and 0 ≤ µ < 1, we have the series expression for f1(z, F) + f − +1 (z, F) +f1(z, F) + f − +1 (z, F) = ωeµπii +(2π)3Q +∞ +� +k,n=1 +µF(k) +kn2 +× 2πi +(2πnkQ2) +1 +2 exp +−3 +4πi + z +2 + e +3 +2 πi−z +4πnkQ2 + +× Γ(2 + µ)Γ(2 − µ)W− 3 +2 , µ + +e +3 +2 πi−z +2πnkQ2 + − +M +� +m1=0 +R(1) +k,n,m1(z, µ) − +M′ +� +m2=0 +R(1) +k,n,m2(z, µ) + ++ ωe−µπi +(2π)3Qi +∞ +� +k,n=1 +µF(k) +kn2 +× 2πi +(2πnkQ2) +1 +2 exp + +3 +4πi + z +2 + e− 3 +2 πi−z +4πnkQ2 + +× Γ(2 + µ)Γ(2 − µ)W− 3 +2 , µ + +e− 3 +2 πi−z +2πnkQ2 + − +M +� +m1=0 +R(2) +k,n,m1(z, µ) − +M′ +� +m2=0 +R(2) +k,n,m2(z, µ) + +− +ωeµπi +(2π)3Qi +∞ +� +k,n=1 +µF(k) +kn2 +× 2πi +� +(2πnkQ2) +1 +2 exp +� +−π +4i + z +2 + +e +π +2 i−z +4πnkQ2 +� +× Γ(2 + µ)Γ(2 − µ)W− 3 +2 , µ +� e +π +2 i−z +2πnkQ2 +� +− +M +� +m1=0 +R(3) +k,n,m1(z, µ) − +M′ +� +m2=0 +R(3) +k,n,m2(z, µ) + ++ ωe−µπi +(2π)3Qi +∞ +� +k,n=1 +µF(k) +kn2 +× 2πi +� +(2πnkQ2) +1 +2 exp +�π +4i + z +2 + e− π +2 i−z +4πnkQ2 +� + +On some analytic. . . +23 +× Γ(2 + µ)Γ(2 − µ)W− 3 +2 , µ +� e− π +2 i−z +2πnkQ2 +� +− +M +� +m1=0 +R(4) +k,n,m1(z, µ) − +M′ +� +m2=0 +R(4) +k,n,m2(z, µ) + . +(10.11) +Since a lemma similar to Lemma 7.1 also holds, the third and the fourth +series on the right hand side in (10.11) are absolutely and uniformly con- +vergent on every compact subset on the whole complex plane. +Next, by the theorem of residues, (4.7) and (9.5) we have +f2(z, F) + f − +2 (z, F) = +� +L +ζ(s − 1) +F(s) eszds − +� +L +ζ(s − 1) +F(s) eszds += −2πi lim +s→2(s − 2)ζ(s − 1) +F(s) ezs += −2πi e2z +F(2). +(10.12) +Finally, by (6.1) and (9.6) +(10.13) +f3(z, F) + f − +3 (z, F) = 0. +Thus, for |y| < π we have +2πi(f (z, F) + f −(z, F)) = 2πi +� 1 +2πi(f1(z, F) + f − +1 (z, F)) − e2z +F(2) +� += 2πiB(z, F), +(10.14) +where +(10.15) +B(z, F) = 1 +2πi(f1(z, F) + f − +1 (z, F)) − e2z +F(2).n +By (10.11), the function f1(z, F)+ f − +1 (z, F) is absolutely and uniformly con- +vergent on every compact subset on the whole complex plane. Hence, the +function B(z, F) is an entire function. Since the function f −(z, F) has a +meromorphic continuation for y < π by Corollary 4.3, the function +f (z, F) = B(z, F) − f −(z, F) +is analytic for all y < π. Since the function f (z, F) is analytic for z ∈ H and +y < π, f (z, F) can be analytically continued to the whole complex plane. +In a similar manner, f −(z, F) can be analytically continued to the whole +complex plane. Therefore, for all z ∈ C we have +(10.16) +f (z, F) + f −(z, F) = B(z, F). +Finally, we prove the functional equation (4.10). We recall the hypothesis +that the coefficient aF(n) in the Dirichlet series of F is real for all n. Hence, +if ρ is a non-trivial zero of F, then so is ρ. For z ∈ H we have +f (z, F) = lim +n→∞ +� +ρ +0 2) alloys, giving rise to a tri-valent or a mixed-valent +situation. At higher coverage (>3 ˚A), preparations at both low and high temperatures +already force extra Eu atoms to diffuse below the completed EuPt2 layer, leading to +similar di- and tri-valent contributions in the Eu 3d XPS spectra, as shown in the top +part of Fig. 3. Increasing the temperature at high coverage enhances the bulk diffusion, +and leads to even stronger Eu3+ emission compared to Eu2+. +3.1.2. Eu intercalation in vicinal hBN/Pt(111) interfaces +After characterizing the Eu +intercalation below the hBN monolayer on Pt(111), we focus on surfaces vicinal to the +Pt(111) plane, investigated with the curved sample sketched in Fig. 4. The negative sign +of the vicinal angle α corresponds to surfaces with A-type steps ({100} microfacets), +and the positive to B-type step arrays ({111} microfacets) [45]. STM images in Fig. 4 +correspond to three representative points of the curved substrate, namely the Pt(223) +position (α=-11.5◦), a low vicinal angle (α=-2.2◦) close to Pt(111), and the Pt(554) +surface (α=5.8◦). Prior to hBN growth, all vicinal surfaces exhibit well-ordered 1D step +arrays, either at low and high vicinal angles [45]. However, the hBN monolayer induces +drastic structural changes, leading to a more complex nanoscale landscape. Close to the +(111) position, large hBN/Pt(111) areas develop, which alternate with densely bunched +steps. +At larger vicinal angles the step bunching process remains, and the surface +becomes a faceted structure. At the (554) position one observes a rather well ordered +structure consisting of (111) terraces and side facets tilted at approx. 6◦. At the (223) + +A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride +8 +Vicinal angle a +[112] +cPt(111) +hBN +Pt(223) +Pt(554) +B-steps +A-steps +-15° +0° +0° +15° +30nm +hBN +a = -11.5° +a = -2.2° +a = 5.8° +hBN ++Eu ++Eu +0 +40 +80 +120 +0 +30 +Length (nm) +h (nm) +0 +0 +20 +100 +200 +80nm ++Eu +0 +0 +15 +200 +400 +(111) +side facets +(111) +side facets +(111) +side facets +40nm +Figure 4. STM images of the (Eu)/hBN/c-Pt(111) system. Large scale STM +images of a hBN monolayer prior and after Eu intercalation collected at three different +positions on the curved c-Pt(111) substrate (scanning parameters: I=0.13 nA; U=0.5 +V). The line scans were taken close to the top of the images corresponding to the Eu +intercalated systems. (111) and side facets are indicated by different color. +position one does not get a clear long range order. The latter situation is similar to +stepped Ni crystals covered by hBN [56], while the ordered structures are rather close +to the Rh case, where hBN growth on stepped surfaces leads to periodically arranged +nanofacets [57]. One interesting question is whether the hBN forms a continuous, defect- +free coat over the hill-and-valley structure underneath, since this requires “bending” of +the hBN layer over Pt substrate facets. +In general terms, however, one expects an +increasing number of defects for an increasing presence of facet/step boundaries at large +vicinal angles. +Eu deposition and intercalation on the vicinal surfaces changes the facet periodicity, +size and inclination, as observed in the STM images. +At the A-step type Pt(223) +position, the rather disordered hBN/Pt(223) structure transforms into a well ordered +array after Eu intercalation. On B-type steps, however, the Eu intercalation is leading +to smaller facets. A statistical analysis of the STM images reveals an increasing average +facet distance [(111)+side facet] of 20 facets/µm close to (111), 25 facets/µm at (223), +and 65 facets/µm at (554). +This means that at A-type steps the surface rugosity +decreases, contrary to B-type steps. On the other hand, XPS spectra (see below) reveal +a similar Eu2+/Eu3+ relation on the different crystal positions, with a slightly higher +di-valent amount at (111) compared to stepped surface planes. +3.2. Magnetism of the hBN/EuPt2 system in the (111) plane +Divalent Eu (4f 7, J = 7/2) has a low-temperature ferromagnetic state in bulk EuPt2 and +similar compounds [58]. We measured the magnetic properties of our hBN-protected + +A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride +9 +1.4 +1.3 +1.2 +1.1 +1.0 +1180 +1160 +1140 +1120 +Photon energy (eV) +-0.2 +-0.1 +0.0 + µ+ + µ− + µ− − µ+ +T = 7 K +q = 70° +2+ +Eu +3+ +Eu +TEY intesnity (arb. un.) +(a) +(b) +1.3 +1.2 +1.1 +1.0 +0.9 +1180 +1160 +1140 +1120 +m H = 6 T +0 +XMCD +Eu M4,5 +2+ +Eu +3+ +Eu +XAS +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +-6 +-4 +-2 +0 +2 +4 +6 +0.02 +0.01 +0.00 +30 +20 +10 +0 +paramagnetic +ferro- +magnetic +m H/M (arb. un.) +0 +2 +M (arb. un.) +XMCD intensity (arb. un.) +Applied field m H (T) +0 +2+ +Eu mag. curve +M4 +M5 +q = 70° +Figure 5. Magnetic properties of intercalated Eu below hBN/Pt(111). (a) +X-ray absorption spectrum (total electron yield - TEY) of horizontally (top, XAS) +and circularly polarized light with opposite sign and its resulting difference spectrum +shown below (XMCD) of 3˚A Eu intercalated below hBN/Pt(111). The XAS spectrum +was fitted with several gaussian profiles (black, yellow, green) and a linear background +(grey) for each of the two main Eu2+ and Eu3+ contributions. (b) Magnetization curve +taken at the maximum of the Eu M5 XMCD signal for a variable applied field from ++6 T to -6 T (blue) and in the opposite direction (orange), respectively. The sample +was oriented with an angle of 70◦ with respect to the applied field at temperature T += 7K. The inset displays the Arrot plot analysis (see text) confirming a ferromagnetic +state. The black line is a linear fit to the high field values. +EuPt2 surface alloy with X-ray circular magnetic dichroism (XMCD) at the (111) +position in the curved crystal. +Results are shown in Fig. 5. +The X-ray absorption +(XAS) spectrum in part (a) reveals a mixture of Eu2+ and Eu3+ contribution after Eu +intercalation below the hBN/Pt(111) surface. This observation confirms the coexistence +of the two Eu valences and it is consistent with the XPS results, namely di-valent Eu +atoms at the EuPt2 surface and tri-valent Eu atoms diffused into the Pt bulk. The +XMCD signal results from the difference of the absorption spectra of left and right +circularly polarized light and is shown in the bottom of Fig. 5(a). At the applied field + +A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride +10 +of 6 T, it shows the typical lineshape of pure di-valent Eu [59]. This is expected since, +as stated before, Eu3+ has a 4f 6 configuration with S = L = 3 and J = 0, hence the +Eu3+ signal should not contribute significantly to the anisotropy. +Ferromagnetism can be probed by measuring the Eu XMCD signal while varying the +applied magnetic field. The XMCD signal is proportional to the magnetization M in the +system. The resulting magnetization curve is shown in Fig. 5(b). It reveals a pronounced +“S” shape. The Arrot plot analysis [60] shown in the inset confirms the ferromagnetic +state of the EuPt2 surface alloy. In such analysis, the square of the magnetization M 2 +is plotted against the applied field divided by the magnetization µ0H/M. A linear fit +of the high field (high µ0H/M) values indicates ferromagnetic state if the line hits the +ordinate at a positive M 2 value, and paramagnetism if the ordinate crossing is negative. +Therefore, the Arrot plot reveals ferromagnetic behavior at the intercalated hBN/Eu- +Pt(111) system. Nevertheless, the magnetization curve shows that both, remanence +and coercive field, are too small to be detectable at the measurement temperature of +7K. For the out-of-plane geometry with the sample normal to the magnetic field and +light incidence, the magnetization curves are slightly more rectangular at small fields +pointing to an out-of-plane easy axis (see supplementary material for details). +3.3. Exposure of the hBN/Eu/Pt system to air +Fig. 6(a) displays the complete evolution of the XPS spectra during the intercalation of +4-˚A-Eu in the hBN/Pt(111) interface and after exposure to air. Prior to Eu intercalation, +we obtain the characteristic shape and positions in the B 1s and N 1s core levels for the +weakly-coupled hBN/Pt(111) system [47]. After Eu intercalation B 1s and N 1s core +levels notably change their shape and energy, reflecting the fact that the hBN contact +interface is now different (EuPt2/hBN), with a stronger hBN-EuPt2 interaction. With +respect to the Eu core level, as mentioned above, it proves that Eu atoms are present +in two configurations, di-valent Eu2+ for the 2D EuPt2 alloy and tri-valent Eu3+ for Eu +atoms incorporated into the Pt bulk below the EuPt2 layer. After sample exposition to +ambient conditions (6 hours, room temperature, 80% humidity) and a vacuum annealing +to 770K to remove impurities from the air exposure, the ratio of Eu2+/Eu3+ drops to +one third. The O 1s core level is detected at the higher binding energy side of the Pt +4p3/2 spectrum. A detailed view (inset) indicates a double-peak with energy positions +of 530.5 eV and 532 eV, respectively. The former is in good agreement with the binding +energy in the tri-valent Eu2O3 oxide [61, 62]. The 532 eV emisssion is interpreted as +due to hydroxide -OH, typical from Eu samples exposed to air. On the other hand, B 1s +and N 1s core levels partially recover their shape and energy prior to Eu exposure. This +indicates that the hBN interaction with partially oxidized EuPt2 patches underneath is +weaker. On the other hand, oxidation of the hBN layer is not observed. +The hBN protection for the intercalated EuPt2 alloy in vicinal Pt substrates is +examined in Fig. 6(b), in direct comparison with the Pt(111) plane. Here we show the +Eu 3d, Pt 4p3/2 and O 1s core levels at the Pt(332) and Pt(223) surfaces. At the Pt(332) + +A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride +11 +4.8 +4.4 +4.0 +3.6 +-196 +-194 +-192 +-190 +-188 +B 1s +16 +14 +12 +10 +8 +PE intensity (counts/sec) +-402 +-400 +-398 +-396 +N 1s +18 +16 +14 +12 +10 +8 +6 +-580 +-560 +-540 +-520 +-500 +8.8 +8.4 +8.0 +7.6 +-538 +-530 + hBN + +4 Å Eu, 870K + +air +12 +10 +8 +6 +4 +-1180 +-1160 +-1140 +-1120 +-1100 +O 1s +Pt 4p3/2 +Eu 3d +N Auger +2+ +Eu +3+ +Eu +14 +12 +10 +8 +6 +-580 +-560 +-540 +-520 +-500 +11 +10 +9 +8 +-538 +-530 +O 1s +Pt 4p3/2 +14 +12 +10 +8 +6 +-1180 +-1160 +-1140 +-1120 +-1100 +Eu 3d +N Auger +2+ +Eu +3+ +Eu + +air +(111) (332) +(223) +Eu/hBN +E-E (eV) +F +(a) +(b) +2+ +Eu +3+ +Eu +2+ +Eu +3+ +Eu +Pt(111) +curved Pt +Figure 6. XPS analysis of Eu intercalation below hBN on the curved Pt +crystal. +(a) N 1s, B 1s, Eu 3d, O 1s, and Pt 4p3/2 X-ray photoemission spectra +taken at hν=1486.6 eV (Al Kα) for the Eu 4 ˚A preparation before and after exposing +to ambient conditions at Pt(111). (b) Comparative Eu 3d, O 1s, and Pt 4p3/2 core +levels at three positions of the curved Pt substrate, at Pt(332), Pt(111) and Pt(335) +positions, respectively. +plane, no di-valent signal reamins in the Eu 3d spectrum after exposure to air. At the +Pt(223) position the Eu2+/Eu3+ is strongly reduced, but the Eu2+ peak is still visible. +Again, the O 1s spectrum suggests that the strong reduction of the number of di-valent +Eu atoms is due to the formation of tri-valent Eu oxides and hydroxides. The overall +O 1s intensity increases as the Eu2+ peak decreases at the stepped surfaces, although a +more detailed peak analysis indicates that the intensity of the Eu-OH peak at 532 eV +is similar in all three cases, and it is the 530.5 eV peak from Eu2O3 the one that scales +reciprocally with the Eu2+/Eu3+ ratio. The complete oxidation of the B-type (332) +surface correlates with the high facet density observed in the STM analysis of Fig. 4, +since this allows a higher number of hBN bending or breaks at facet borders. The rather +small divalent signal that remains in Pt(223) may reflect the presence of larger (111) +and side facets where the intercalated EuPt2 alloy remains better protected at ambient +conditions. +Finally, we analyze the impact of the air exposure on the hBN protecting layer. +For this purpose angle-resolved photoemission spectroscopy (ARPES) is particularly +appropriate, since it is highly sensitive to the hBN valence band, as shown in Fig. 7. +The photoemission intensity map of Fig. 7(a) corresponds to the hBN/Pt(111) interface +(centre of the curved sample), and has been measured using the He II excitation energy +(hν = 40.8eV). The strongly dispersing, intense features are the hBN π bands, with + +A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride +12 +-12 +-10 +-8 +-6 +-4 +-2 +0 +2.0 +1.0 +0.0 +2.0 +1.0 +2.0 +1.0 +-12 +-10 +-8 +-6 +-4 +-2 +0 +0.0 +2.0 +1.0 +hBN +1 Å Eu ++5min. air +annealing +E-E (eV) +F +-1 +|k | (Å ) +|| +(a) +(b) +(d) +(c) +G +K +K +G +K +K +hBN/EuPt2 +hBN/Pt +hBN/EuPt2 +Figure 7. Angle resolved Photoemission analysis of the hBN layer. He IIa +(hν = 40.8eV) photoemission intensity maps along ¯Γ ¯K direction of the hBN band +structure for (a) hBN/Pt(111), (b) after intercalation of 1 ˚A of Eu at T = 770K, (c) +exposure of that sample to air, and (d) after another annealing to 770K to desorb air +contamination, respectively. The most intense features correspond to the π-band of +hBN at the indicated interfaces. +minimum at the ¯Γ point of the Brillouin zone and maximum at ¯K. +The emission +between approx. +2 and 8 eV binding energy entirely belongs to hBN, while the Pt +valence band features appear closer to the Fermi level. After intercalation of a very +small amount of Eu (1 ˚A), the main π band appears unaltered, although a replica of +this band emerges below, at 2 eV higher binding energy (Fig. 7(b)). The unaltered +band corresponds to pure Pt areas without Eu while the shifted band arises in areas +where the Eu intercalates. The band shift is explained by the enhanced interaction of +the hBN with the EuPt2 substrate, with a net electron transfer from Eu atoms to the +hBN layer. After air exposure all bands get quite blurry, due to surface contamination. +The dominating hBN π band is still visible, slightly shifted to higher binding energies. +Interestingly, both the pure hBN/Pt and the Eu-intercalated bands can be recovered by +annealing the sample again to 770K, which removes adsorbates from the surface. This +indicates that the oxygen intercalation does not affect (oxidize) the hBN layer, which +remains intact. The photoemission intensity mapping still includes the higher binding +energy replica, indicating that strongly interacting hBN/EuPt2 patches remain, despite +the partial oxidation of the EuPt2 layer. +4. Conclusions +We have investigated the structural, magnetic and electronic properties of Eu after +intercalation between a Pt substrate and a hBN monolayer. +We used a Pt sample +curved around the (111) direction in order to additionally assess the role of substrate +steps. Our LEED analysis of the (111) interface shows a ∼( +√ +3 × +√ +3)R30◦ pattern, +revealing the presence of the EuPt2 surface alloy under the hBN layer. We find that Eu + +A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride +13 +atoms in this EuPt2 layer are divalent, while Eu atoms that have diffused further inside +the Pt bulk during the intercalation process are trivalent. Interestingly, the Eu2+/Eu3+ +ratio is not affected by the presence of steps at the Pt substrate. XMCD magnetization +curves on the di-valent Eu atom reveal a ferromagnetic behavior. Air exposure of the +sample leads to a partial protection of the magnetic, divalent Eu atoms at the (111) +plane, while at vicinal surfaces the protecting role of the hBN layer is less efficient, as +reflected in the larger attenuation of the divalent Eu state. Such incomplete protection +of vicinal planes may be related to a larger number of defects and domain boundaries +in a more discontinuous hBN layer, since this covers a much rougher hill-and-valley +faceted structure. This facilitates oxygen diffusion, intercalation and the EuPt2 alloy +oxidation. In contrast, the hBN layer itself remains intact upon both Eu intercalation +and air exposure. +Supplementary material +Supplementary material contains information on electron spectroscopy analysis for Eu +intercalation at insufficient substrate temperatures. Furthermore, the magnetic easy +axis direction is investigated by XMCD technique. +Acknowledgments +We acknowledge financial support from grants PID2020-116093RB-C44 funded by the +Spanish MCIN/AEI/ 10.13039/501100011033 and the Basque Government (Grant IT- +1591-22). We acknowledge the European Synchrotron Radiation Facility for provision +of beam time on ID32. ESRF access was provided through proposal MA-5454 [63]. Part +of the research leading to the result has been supported by the project CALIPSOplus +under Grant Agreement 730872 from the EU Framework Programme for Research +and Innovation HORIZON 2020. Y. 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Rev. Research 4 013237 +[60] Arrott A 1957 Criterion for ferromagnetism from observations of magnetic isotherms Phys. Rev. +108 1394–1396 +[61] Mercier F, Alliot C, Bion L, Thromat N and Toulhoat P 2006 XPS study of Eu(III) coordination +compounds: Core levels binding energies in solid mixed-oxo-compounds EumXxOy Journal of +Electron Spectroscopy and Related Phenomena 150(1) 21–26 +[62] Baltrus J P and Keller M J 2019 Rare earth oxides Eu2O3 and Nd2O3 analyzed by XPS Surface +Science Spectra 26(1) 014001 +[63] Mohammed Idris Bakhit A, Schiller F, Fernandez Gomez-Recuero L, Castrillo Bodero R and +Hasegawa Y 2022 Magnetic properties in the single molecular magnet TbPc2 insulating +ferromagnet EuO2 system https://doi.org/10.15151/ESRF-ES-886076657 + +Supplementary material of “A ferromagnetic Eu-Pt +surface compound grown below hexagonal boron +nitride” +Alaa Mohammed Idris Bakhit1,2, Khadiza Ali3, 4, Anna A. +Makarova5, Igor P´ıˇs6, Federica Bondino6, Roberto Sant7, Saroj +P. Dash4, Rodrigo Castrillo1, Yuri Hasegawa1,8, J. Enrique +Ortega1,3,9, Laura Fernandez1, and Frederik Schiller1,3 +1 Centro de F´ısica de Materiales CSIC-UPV/EHU-Materials Physics Center, +E-20018 San Sebasti´an, Spain +2 Departamento de Pol´ımeros y Materiales Avanzados: F´ısica, Qu´ımica y +Tecnolog´ıa, Universidad del Pa´ıs Vasco UPV/EHU, 20018 San Sebasti´an, Spain +3 Donostia International Physics Center, E-20018 Donostia-San Sebasti´an, Spain +4 Chalmers University of Technology, G¨oteborg, Chalmersplatsen 4, 412 96 G¨oteborg, +Sweden +5 Physikalische Chemie, Institut f¨ur Chemie und Biochemie, Freie Universit¨at Berlin, +Arnimallee 22, 14195 Berlin, Germany +6 IOM-CNR, Strada Statale 14 Km 163.5, I-34149 Trieste, Italy +7 ESRF, The European Synchrotron, 71 Avenue des Martyrs, CS40220, 38043 +Grenoble Cedex 9, France +8 Department of Physical Sciences, Ritsumeikan University, Kusatsu, 525-8577, Japan +9 Departamento de F´ısica Aplicada I, Universidad del Pa´ıs Vasco UPV/EHU, +E-20018 San Sebasti´an, Spain +E-mail: frederikmichael.schiller@ehu.es +The supplementary information file contains: +Supplementary Figures S1-S4 +Supplementary information and data are provided to further illustrate the +intercalation process of Eu below the hBN/Pt substrate as well as magnetic easy axis +determination of the Eu-Pt alloy below hBN. +arXiv:2301.11837v1 [cond-mat.mtrl-sci] 27 Jan 2023 + +Supplementary material +S2 +S1. Spectroscopy analysis of insufficient temperature for Eu intercalation +Eu does not completely intercalate below a hBN layer on Pt if the substrate temperature +is not sufficient high. +Here we will show Near-edge X-ray absorption fine structure +(NEXAFS) and X-ray photoelectron spectroscopy (XPS) data for such preparations. +Data were taken at ID32 and BACH beamlines of ESRF and Elettra synchrotrons, +respectively. +X-ray absorption spectroscopy (normal to the field and light incidence) for a approx. +1 ML thick Eu film deposited at a T = 610K hot hBN/Pt(111) substrate and successive +exposure to air is shown in Fig. S1. The air exposure was carried out at with the sample +at room temperature during five minutes. After air exposure a strong transformation +of the spectral shape towards tri-valent Eu configuration took place. Nevertheless, even +after the air exposure experiment a small di-valent part is still present. In order to +remove remaining air contamination, the sample was post-annealed in UHV to T = +470K during 10 minutes. +Fig. S2 represents X-ray photoemission data from BACH beamline taken at a +photon energy hν = 272eV. The substrate temperature during Eu deposition for the data +shown in Fig. S2 was T = 470K. The data set includes measurements of hBN/Pt(111) +and two successive Eu intercalations, each for 10 min at a very low rate. The total +Eu thickness was calculated from the Pt 4f intensity loss taking into account the mean +free path of the electrons. The latter was extracted from the universal curve of the +electron mean free path [1] and resulted in Eu thickness of 0.7 and 1.5˚A for 10 and +20 min Eu deposition, respectively. In Fig. S2 one observes the spin orbit split 4f7/2 +and 4f5/2 components of the Pt 4f core level. These are further split in the surface +(S) and bulk (B) emissions indicated in the Figure. It is interesting to note that the +surface emission can be still observed for hBN/Pt(111). This is possible due to the +weak interaction of hBN and the Pt, specially at the (111) face that hosts a hBN Moire +1.3 +1.2 +1.1 +1.0 +TEY intensity (arb. un.) +1190 +1180 +1170 +1160 +1150 +1140 +1130 +1120 +Photon energy hn (eV) +T = 300 K +q = 0° +Eu M4,5 + Eu/hBN/Pt(111) + +Air exposure +2+ +Eu +3+ +Eu +M5 +M4 +Figure S1. X-ray absorption spectra at the Eu M4,5 absorption edge for Eu +deposition on hBN/Pt(111) at T = 610K and successive air exposure. + +Supplementary material +S3 +PE intensity (arb. un.) +-200 +-150 +-100 +-50 +0 +E-E (eV) +F + hBN + 10 min Eu @ 470K + 20 min Eu @ 470K +hn = 272 eV +B 1s +Pt 4f +-194 +-192 +-190 +-188 + A = 0.93 AhBN ® 0.35 Å Eu + A = 0.83 AhBN ® 0.9 Å Eu + AhBN +B 1s +-76 +-74 +-72 +-70 + pure Pt(111) hn = 190eV + A = 0.95 AhBN ® 0.7 Å Eu + A = 0.87 AhBN ® 1.5 Å Eu + AhBN/Pt(111) +Pt 4f +S +B +S +B +4f7/2 +4f5/2 +Figure S2. XPS analysis for sub-monolayer Eu deposition on hBN/Pt(111) +at T = 470K. XPS survey, B 1s and Pt 4f emissions of 0.7 and 1.5˚A Eu coverage +on hBN/Pt(111), respectively, taken for hν = 272eV. In the case of the Pt 4f also the +clean Pt emission is included to better observe the surface and bulk contributions (hν += 190eV). After low binding energy normalization, the areas A below the emissions +were extracted and from them the overlayer amount was calculated by taking into +account the electron mean free path of the electrons from the standard curve [1]. From +this analysis we observe that not all Eu is intercalated below the hBN layer but rather +stay on top. +lattice. After Eu deposition the surface component shrinks much more than the bulk +component, revealing that partial Eu intercalation occurred. We again calculate the Eu +overlayer thickness but now from the B 1s core level that give us the amount of Eu on +top of the hBN layer. The thickness that results from the area diminution is 0.35 and +0.9˚A, respectively, for the two evaporations. This means that during the first deposition +0.35˚A intercalated and after the second deposition 0.6˚A out of the 1.5˚A Eu intercalated. +We also observe the doping effect of Eu towards the hBN layer that shifts the B 1s core +level to higher binding energies in a similar way as for complete intercalation, see Fig. +6(a) of the main text. +The analysis of the valency has been carried out with resonant photoemission +applying photon energies around the 4d→4f absorption edge. +The results can be +found in Fig. S3. +At off-resonant photoemission (hν = 132eV) the Eu 4f emission +is strongly suppressed, therefore the valence band is dominated by Pt 5d valence +band emission. Nevertheless, there is already some Eu 4f emission. The 4f emissions +are much better observed for the resonant photon energies hν = 140eV and 144.5eV +corresponding to Eu2+ and Eu3+ resonances, respectively [2, 3, 4]. The di-valent emission +is located at approx. 1.1eV binding energy, the tri-valent multiplet is observed between +4 and 12eV. Due to the unknown cross-section effects around the resonances an exact +Eu2+/Eu3+ ratio cannot be extracted. The tri-valent contribution arises from a part + +Supplementary material +S4 +300 +250 +200 +150 +100 +50 +0 +-30 +-25 +-20 +-15 +-10 +-5 +0 +on-resonant +2+ +Eu +200 +150 +100 +50 +0 +-30 +-25 +-20 +-15 +-10 +-5 +0 +60 +40 +20 +0 +-30 +-25 +-20 +-15 +-10 +-5 +0 +off-resonant +2+ +Eus +EuO +3+ +Eu +4f +Eu 5p +hn = 132eV +hn = 140eV +hn = 144.5eV +on-resonant +3+ +Eu +1.5Å Eu @ 470K ++air exposure +3 +PE intensity (10 counts) +E-E (eV) +F +Pt 5d+ +Eu 4f +Figure S3. +Resonant Photoemission at the Eu 4d→4f absorption edge. +Normal emission photoemission spectra of the valence band region of 1.5˚A Eu deposited +onto a 470K hot hBN/Pt(111) surface for three photon energies corresponding to off, +on-resonant Eu2+, and on-resonant Eu3+ at hν = 132, 140, and 144.5eV, respectively. +Shown are spectra prior and after exposure of the sample to ambient conditions (5 +min, room temperature). + +Supplementary material +S5 +of the intercalated Eu atoms that diffuse further inside the Pt bulk, but can also arise +from oxidation of Eu to tri-valent Eu in Eu2O3 of the Eu atoms atop of the hBN layer. +The latter oxidation is very difficult to avoid due to the strong reactivity of Eu even +under very good ultra-high vacuum conditions [5]. After air exposure, the situation +changes drastically. There is still a small di-valent signal, but now at a binding energy +of 2.3eV. This Eu2+ emission cannot arise from di-valent Eu surrounded by a metallic +environment whose binding energy is always lower than 2eV. But an emission at such +2.3eV arises from divalent Eu in EuO at surfaces [6, 3]. This oxide is usually unstable +under ambient conditions but seems to be able to form and remain stable under used +experimental conditions. EuO is created either on the surface or at the interface under +hBN (and stabilized thanks to the interface). Further investigations would be necessary +to distinguish both possibilities. In any case, we do not observe a possible divalent +EuPt2 structure below the hBN after the air exposure process for Eu deposition at +470K substrate temperature. +S2. Magnetic anisotropy determination +The magnetic anisotropy of the intercalated Eu can be determined from the two different +geometries applied. The magnetization curves for the sample perpendicular and at an +angle of 70◦ with respect to the applied field is shown in Fig. S4. The close-up at small +fields reveal a more rectangular shape of the out-of-plane geometry magnetization loop +pointing to this direction as the easy axis of magnetization. The XMCD values close to +zero-field has been taken from individual complete XMCD measurements to avoid the +typical zero-field artifacts of XMCD measurements. +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +XMCD signal (arb. un.) +-6 +-4 +-2 +0 +2 +4 +6 +Applied field m H (T) +0 +-0.05 +0.00 +0.05 +-1 +0 +1 + θ = 0° + θ = 70° + + + θ = 0° + + θ = 70° +Applied field m H (T) +0 +XMCD (arb. un.) +2+ +Eu mag. curve +Figure S4. Eu magnetization curves at different geometries. Eu magnetization +curves normal and at θ = 70◦ with respect to the field. The inset shows a closeup +at small fields to distinguish better the two curves. One clearly observes the more +rectangular like shape for the out-of-plane geometry revealing an out-of-plane easy +axis. + +Supplementary material +S6 +[1] Dench W A and Seah M P 1979 Quantitative Electron Spectroscopy of Surfaces: A Standard Data +Base for Electron Inelastic Mean Free Path in Solids Surf. Interf. Anal. 1 1 +[2] Schneider W D, Laubschat C, Kalkowski G, Haase J and Puschmann A 1983 Surface effects in eu +intermetallics: A resonant photoemission study Phys. Rev. B 28 2017–2022 +[3] Santana J A C, Liu P, Wang X, Tang J, McHale S R, Wooten D, McClory J W, Petrosky J C, +Wu J, Palai R, Losovjy Y B and Dowben P A 2012 The local metallicity of gadolinium doped +compound semiconductors Journal of Physics: Condensed Matter 24(44) 445801 +[4] Banik S, Bendounan A, Thamizhavel A, Arya A, Risterucci P, Sirotti F, Sinha A K, Dhar S K +and Deb S K 2012 Electronic structure of eucu2ge2 studied by resonant photoemission and x-ray +absorption spectroscopy Phys. Rev. B 86 085134 +[5] Schumacher S, Huttmann F, Petrovi´c M, Witt C, F¨orster D F, Vo-Van C, Coraux J, Mart´ınez- +Galera A J, Sessi V, Vergara I, R¨uckamp R, Gr¨uninger M, Schleheck N, Meyer zu Heringdorf +F, Ohresser P, Kralj M, Wehling T O and Michely T 2014 Europium underneath graphene on +Ir(111): Intercalation mechanism, magnetism, and band structure Phys. Rev. B 90 235437 +[6] Caspers C, M¨uller M, Gray A X, Kaiser A M, Gloskovskii A, Fadley C S, Drube W and Schneider +C M 2011 Chemical stability of the magnetic oxide euo directly on silicon observed by hard x-ray +photoemission spectroscopy Phys. Rev. B 84 205217 + diff --git a/QNFKT4oBgHgl3EQfhS5q/content/tmp_files/load_file.txt b/QNFKT4oBgHgl3EQfhS5q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..377ff74f69a92c0140e06f39bed232880d6bd8da --- /dev/null +++ b/QNFKT4oBgHgl3EQfhS5q/content/tmp_files/load_file.txt @@ -0,0 +1,930 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf,len=929 +page_content='A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride Alaa Mohammed Idris Bakhit1,2, Khadiza Ali3,4, Anna A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Makarova5, Igor P´ıˇs6, Federica Bondino6, Roberto Sant7, Saroj P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Dash4, Rodrigo Castrillo1, Yuri Hasegawa1,8, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Enrique Ortega1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Laura Fernandez1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' and Frederik Schiller1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='3 1 Centro de F´ısica de Materiales CSIC-UPV/EHU-Materials Physics Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' E-20018 San Sebasti´an,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Spain 2 Departamento de Pol´ımeros y Materiales Avanzados: F´ısica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Qu´ımica y Tecnolog´ıa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Universidad del Pa´ıs Vasco UPV/EHU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 20018 San Sebasti´an,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Spain 3 Donostia International Physics Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' E-20018 Donostia-San Sebasti´an,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Spain 4 Chalmers University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' G¨oteborg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Chalmersplatsen 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 412 96 G¨oteborg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Sweden 5 Physikalische Chemie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Institut f¨ur Chemie und Biochemie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Freie Universit¨at Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Arnimallee 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 14195 Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Germany 6 IOM-CNR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Strada Statale 14 Km 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5, I-34149 Trieste, Italy 7 ESRF, The European Synchrotron, 71 Avenue des Martyrs, CS40220, 38043 Grenoble Cedex 9, France 8 Department of Physical Sciences, Ritsumeikan University, Kusatsu, 525-8577, Japan 9 Departamento de F´ısica Aplicada I, Universidad del Pa´ıs Vasco UPV/EHU, E-20018 San Sebasti´an, Spain E-mail: frederikmichael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='schiller@ehu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='es Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' One of the fundamental applications for monolayer-thick 2D materials is their use as protective layers of metal surfaces and in-situ intercalated reactive materials in ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Here we investigate the structural, electronic, and magnetic properties, as well as the chemical stability in air of a very reactive metal, Europium, after intercalation between a hexagonal boron nitride (hBN) layer and a Pt substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' We demonstrate that Eu intercalation leads to a hBN-protected ferromagnetic EuPt2 surface alloy with divalent Eu2+ atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' We expose the system to ambient conditions and find a partial shielding of the Eu-Pt interface, which remains ferromagnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The use of a curved Pt substrate allows us to explore ferromagnetism and the ambient pressure protection with other substrate planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The interfacial EuPt2 surface alloy formation remains the same, but the resistance to ambient conditions is reduced, likely due to a rougher surface and a more discontinuous hBN coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Keywords: ambient condition protection, 2D Materials, ferromagnetic surface alloy formation arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='11837v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='mtrl-sci] 27 Jan 2023 A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Introduction Ferromagnetic two-dimensional (2D) structures are of uttermost importance in spintronics applications [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Such 2D magnetic systems can either be 2D Van der Waals (vdW) ferromagnets [2, 3], or ultrathin magnetic overlayers [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Both types of systems present quantum and topological phases [5], but achieving new exotic properties requires design and investigation of novel materials and architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' In thin magnetic overlayers, the surface contains transition and/or rare-earth metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The interest in such 2D ferromagnetic systems is prompted by the reduced atomic-scale size and the diversity of magnetic states that arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The synthesis of 2D vdW ferromagnets is significantly different compared to the direct epitaxy of magnetic layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 2D vdW systems are exfoliated from crystals, which in turn are grown by ex-situ chemical vapor or atomic layer deposition [6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Magnetic overlayers, however, are in-situ grown by physical vapor deposition in vacuum, a process that does not guarantee a sharp 2D system that stands harsh conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Achieving the latter demands a detailed in-situ, atomic-scale investigation, comprising structure and electronic states, as well as chemical stability in ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 2D magnetic alloys are quite reactive in air, loosing or changing their magnetic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Protection of such surfaces can be achieved by means of ceramic coatings, polymer protection films or deposition of non-reactive metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Nevertheless, most of these protecting overlayers influence the magnetism of the surfaces leading to a variation of the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Recently, protection of the surfaces by graphene (Gr) [9, 10, 11, 12, 13, 14, 15, 16, 17], hexagonal boron nitride (hBN) [10, 18, 19, 20, 21, 22, 23] and a mixture of both materials [24] is being considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' In this context, the use of a hBN protecting layer is very appealing, since it would provide close contact of the ferromagnetic material with a wide-gap semiconductor, enabling charge injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Therefore, the question that arises is whether we can achieve a sufficiently protective hBN layer that preserves the magnetic properties of the 2D compound in ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Here, we study a hBN-protected ferromagnetic Eu-Pt surface alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The Eu-Pt compound is formed after Eu intercalation under the hBN film previously grown on a Pt crystal surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Metal intercalation below Gr or hBN overlayer has been extensively studied over the last two decades [15, 16, 17, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The purpose in the majority of the works was to separate the 2D overlayer from the substrate [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Most often this is done by the intercalation of noble metal atoms like Au, Ag, or Cu [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Conversely, in order to force a stronger 2D material interface interaction, one can proceed with the intercalation of alkaline [29] or earth-alkaline metals [30, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' If a too-strongly interacting substrate-2D overlayer is achieved, additional intercalation of oxygen lifts again the 2D layer and re-establishs the original 2D material properties [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' However, oxygen exposure may result in the oxidation of the protecting hBN layer [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Eu intercalation has been less investigated [25, 34, 35, 17, 16] despite of its interesting magnetic properties, mainly due to the strong reactivity of this rare earth metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' All Eu intercalation studies have been carried out on graphite or graphene epilayers, but A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride 3 no experiments exist using hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Among the different rare earth metal compounds, europium alloys are particularly interesting due to the various valence states of the Eu atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' It may adopt a di-valent Eu2+, a tri-valent Eu3+, or even a mixed-valent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' For trivalent Eu3+, Eu has a 4f 6 configuration with S = L = 3 and J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The ground state J-multiplet level (in 2S+1LJ configuration) is 7F0 presenting a non-magnetic singlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This situation differs from divalent Eu2+ with 4f 7 configuration, S = 7/2, L =0 and J = 7/2 leading to a 8S7/2 ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' In the latter case, Eu2+ is able to form ferromagnetic compounds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=', europium chalcogenides [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The different valence states of Eu are found to depend on several factors: the surrounding material, the lattice pressure, the number and type of nearest neighbors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' In the particular case of Eu-Pt compounds there is a smooth valence transition when changing the stoichiometry from EuPt5 (completely tri-valent) to EuPt2 (Eu atoms in a di-valent state) [37, 38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Additionally, valence instabilities can be induced in EuPt3 by high pressure [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Valence changes may also happen at the surface due to a reduced coordination [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Such transitions from trivalent to divalent configuration have also been observed for Eu-Ni or Eu-Pd compounds [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Here we present a Eu-Pt surface alloy formed after intercalation of Eu at the hBN/Pt interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' First, we perform a structural analysis of the interface, followed by a detailed electronic and magnetic characterization of the Eu-Pt compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' We demonstrate that the topmost layer under the hBN coat is a 2D EuPt2 surface alloy, with di-valent Eu atoms that reveal ferromagnetic behavior at low temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Next, we check the efficiency of the hBN layer protection in ambient pressure, by analyzing the electronic properties prior and after air exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The sample is a platinum crystal curved around the (111) direction (c-Pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This provides a smooth variation of the crystallographic orientation across the (macroscopic) surface, allowing us to extend the analysis of the EuPt2 alloy to vicinal Pt crystal planes, characterized by a high density of atomic steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' By scanning our different experimental (electron,photon) probes on top, we can rigorously study the influence of steps and terraces in the structural, magnetic, and electronic properties of the EuPt2 surface alloy, as well as the protecting quality of the hBN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Experimental details The growth and electronic properties were mainly investigated at the Nanophysics laboratory in San Sebastian, Spain using a combined system containing scanning tunneling microscopy (STM), low energy electron diffraction (LEED), X-ray photoemission (XPS) and angle-resolved photoemission spectroscopy (ARPES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Part of the electronic structure investigations have been carried out at BACH beamline of Elettra synchrotron (Trieste, Italy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The XPS setup in the laboratory is equipped with a Specs Al Kα µ-FOCUS 600 monochromator while the ultraviolet light source consists of a Specs UVS-300 discharge lamp with monochromator (Specs TMM 304) tuned to HeIIα light with hν = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='8eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' At Elettra synchrotron, p-polarized light was applied A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride 4 at a photon energy of 272eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' All measurements were taken with the sample at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' STM experiments were carried out in a Omicron VT-setup by holding the sample at room temperature and scanning with a W tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The analysis of the STM images has been performed with WSXM software [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The magnetic properties were investigated at ID 32 of the European synchrotron radiation facility (ESRF) by means of X-ray magnetic circular dichroism (XMCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' For this purpose the sample was placed normal or grazing (70◦) with respect to the incoming photon beam and field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The field was ramped between +6 and -6 T with the sample hold at T = 7K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Horizontal, left and right circularly polarized light (99% polarization) was used for photon energies around the Eu M4,5 X-ray absorption edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' As a substrate material, a cylindrical sector of a Pt (c-Pt) single crystal was used whose cylinder axis is along a [1¯10] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The centre of the curved surface points towards the [111] direction, while the borders are oriented ±15◦ with respect to the (111) center (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This curved Pt surface was cleaned by Ar ion sputtering (room temperature) and temperature annealing (1000K) as well as by occasional oxygen heating (2×10−8 mbar O2, 950K) followed by a flash in UHV to 1050K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This standard procedure, as described elsewhere [45, 46], leads to sharp LEED patterns where the typical step splitting was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' hBN was grown by chemical vapor deposition (CVD) process from borazine precursor (KATCHEM spol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' s r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' For this purpose the curved Pt crystal was held at 1020K while borazine was dosed for 20 minutes at 2×10−7 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' As can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 1, this growth produces a sharp and well ordered Moire pattern in LEED at the Pt(111) position of the curved crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' On the other substrate positions, corresponding to the vicinal surfaces in the mentioned ±15◦ range around (111), the LEED reveals less ordered structures with line like features pointing to a multi-facet structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Eu was deposited in a third step on top of this hBN/c-Pt substrate while the sample was held at an elevated temperature to allow Eu intercalation below hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' As pointed out earlier [25], the high temperature is quite important to immediately protect the Eu from oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' We used substrate temperatures between 570 and 870K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The deposition process was carried out in UHV systems with a base pressure prior to Eu deposition below 1×10−9 mbar not surpassing 3×10−9 mbar during deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' For lower substrate temperatures incomplete intercalation takes place and part of the Eu stays on top of the hBN, see supplementary material for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The oxidation protection experiments consisted in exposing the sample to ambient pressure conditions (6 hours, room temperature, 80% humidity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Results and Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Formation of Eu-Pt surface alloy below hBN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Eu intercalation in the hBN/Pt(111) interface The structural evolution of the hBN/Eu/Pt intercalated system can readily be monitored with low energy electron A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride 5 diffraction (LEED) experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' LEED patterns in Figure 1 correspond to the (111) plane on the Pt curved crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The clean Pt(111) pattern is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 1(a), which transforms, after Borazine dosing at T = 1020K, into the characteristic hBN (9×9) Moire of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 1(b) [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' For successful Eu intercalation, there is a threshold temperature of the substrate of T = 770K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Below this temperature the majority of the rare-earth material stays on top of the hBN coat, being subject to rapid contamination/oxidation (see Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' At the lower range of the intercalation temperatures, right above 770K, the LEED pattern only shows the progressive extinction of the hBN Moire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' When rising the temperature to T = 870K a new ≈( √ 3 × √ 3) R30◦ pattern emerges [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 1(c)], with some weak satellite spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' A detailed inspection of the ≈( √ 3 × √ 3) structure reveals a 10/11·( √ 3 × √ 3) R30◦ geometry with respect to Pt(111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This pattern reflects the presence of a EuPt2/Pt(111) Moire-like coincidence lattice, similar to those found in rare earth RE-Au and RE-Ag surface alloys with RE-Au2 and RE-Ag2 composition [48, 49, 50, 51, 52, 53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The ( √ 3 × √ 3) ordering arises from the 1:2 Pt:Eu stoichiometry of the alloy at the local atomic-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The Moire emerges from the lattice mismatch of the 2D RE-noble metal alloy layer and the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The hBN/Eu/Pt(111) system is different from the Gr/Eu/Ir(111) one [25], where the superstructure LEED spots belong to the graphene diffracted beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' In that case it was proposed that Eu forms a floating layer between the Ir(111) substrate and the graphene layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' In the here considered hBN/Eu/Pt system, however, the strongest LEED spots correspond to the EuPt2 layer at the Pt interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' As indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 1(c), we can still detect extra satellite spots around the 10/11·( √ 3× √ 3) R30◦ structure, which arise from the coincidence lattice defined by the mismatched hBN/EuPt2 interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Being all LEED structures properly identified, one can calculate real space lattice parameters out of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Taking into account the Pt lattice constant of aPt = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='92˚A, we obtain the EuPt2 coincidence lattice constant of 11·aPt/ √ 2 = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5˚A, from which we deduce the EuPt2 lattice parameter aEuPt2 = aPt/ √ 2·10/11· √ 3 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='29˚A , with a nearest neighbor distance of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='05˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The lattice mismatch of the EuPt2 layer with the +borazine, 1020K +Eu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 870K +air;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' +anneal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 570K Pt(111) (a) (c) (d) 10 11 (Ö3´Ö3)R30° (b) (1´1) Moire hBN/Pt Moire EuPt /Pt 2 hBN/EuPt2 EuPt /Pt 2 (9´9) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' LEED images along the preparation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' (a): Pt(111), (b) after borazine exposure at 1020K producing a (9×9) Moire pattern, (c) additional Eu deposition/intercalation at Tsample = 870 K leading to EuPt2 layer below hBN and a different Moire pattern, (d) after 6 hours room temperature air exposure and subsequent 570 K vacuum annealing (LEED kinetic energy 70eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride 6 (a) EuPt2 Laves phase C15 (b) Kagomé (111) Pt neighbor (111) Eu (c) ( 3 × 3) R30° EuPt2 surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' alloy Eu Pt Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Structural Model of the EuPt2 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' (a): Laves phase (C15) of bulk EuPt2 in the MgCu2 structure, (b) central Pt Kagom´e (blue layer in (a)) and neighboring Eu (111) layer stacking inside bulk EuPt2 that would result in a PtEu3 composition with (2×2) superstructure after Eu incorporation, (c) ( √ 3 × √ 3) R30◦ monolayer surface alloy structure below a 2D hBN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' hBN lattice on top (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='504˚A) is quite large, but explains the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='6×4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='6 weak superstructure spots, marked by blue arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Bulk EuPt2 exists and crystalizes in a MgCu2 Laves phase structure, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' A bulk lattice constant aLaves between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='64 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='73 ˚A has been reported [37, 55, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' However, in such bulk EuPt2 structure, and along the [111] direction, one cannot find any stoichiometric EuPt2 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Contiguous (111) planes contain either Pt and Eu solely, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The plane containing the Eu atoms is shifted by 3/8 of the densely packed Pt planes (approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='8 ˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The fundamental Pt containing (111) plane is formed by a Kagom´e lattice [blue layer in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 2(a)], with the Eu atoms sitting above and below each Pt hexagon of the Kagom´e lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Considering the Eu-Pt bilayer, this defines a (2×2) superstructure with a EuPt3 composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The here found EuPt2 structure formed below hBN is therefore not related to bulk EuPt2 and can be understood by the simple incorporation of Eu atoms in the uppermost Pt(111) surface plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Due to the larger size of the Eu atom, the interatomic distance at the surface increases, and a mismatch with the Pt(111) substrate underneath arises, leading to the 10/11·( √ 3 × √ 3) R30◦ coincidence lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The chemical characterization of the Eu intercalation process is carried out by means of X-ray photoemission spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' In the bulk EuPt2 compound Eu atoms are in a di-valent Eu2+ configuration, while Eu atoms in compounds with higher Pt content become mixed-valent or completely tri-valent [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The latter situation would be the case for Eu interstitial atoms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=', those that diffuse into the bulk and are surrounded by Pt completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 3 reveals the Eu 3d core level for sub- and monolayer preparations at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Submonolayer Eu deposition and intercalation at low substrate temperature (T = 570K) leads to a (nearly) complete divalent configuration while preparations at higher T result in the appearance of an additional Eu3+ signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' We interpret these observations as follows: at lower temperature only a partial Eu intercalation takes place, leading to the formation of A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride 7 Photoemission Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=') 1180 1160 1140 1120 1100 E-EF (eV) Eu 3d N Auger Eu 2+ Eu 3+ Sub-ML preparation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 Å, T = 870K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5 Å, T = 570K ML-preparation 4 Å, T = 870K 3 Å, T = 570K Eu 3+ Eu 2+ 3d5/2 3d3/2 Eu 3+ sat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Eu 3d photoemission signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' X-ray photoemission spectra for the Eu 3d edge taken at hν = 1486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='6eV (Al Kα) for sub- and monolayer preparations at sample temperatures T = 570K and 870K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' EuPt2 patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' For higher temperature, the Eu intercalation is complete, but together with the Eu2+ species at the EuPt2 interface the Eu3+ component arises, which can be ascribed to Eu interstitials in the Pt bulk, or simply to the buildup, under the topmost EuPtn patches, of EuPtn (n > 2) alloys, giving rise to a tri-valent or a mixed-valent situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' At higher coverage (>3 ˚A), preparations at both low and high temperatures already force extra Eu atoms to diffuse below the completed EuPt2 layer, leading to similar di- and tri-valent contributions in the Eu 3d XPS spectra, as shown in the top part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Increasing the temperature at high coverage enhances the bulk diffusion, and leads to even stronger Eu3+ emission compared to Eu2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Eu intercalation in vicinal hBN/Pt(111) interfaces After characterizing the Eu intercalation below the hBN monolayer on Pt(111), we focus on surfaces vicinal to the Pt(111) plane, investigated with the curved sample sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The negative sign of the vicinal angle α corresponds to surfaces with A-type steps ({100} microfacets), and the positive to B-type step arrays ({111} microfacets) [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' STM images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 4 correspond to three representative points of the curved substrate, namely the Pt(223) position (α=-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5◦), a low vicinal angle (α=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='2◦) close to Pt(111), and the Pt(554) surface (α=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='8◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Prior to hBN growth, all vicinal surfaces exhibit well-ordered 1D step arrays, either at low and high vicinal angles [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' However, the hBN monolayer induces drastic structural changes, leading to a more complex nanoscale landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Close to the (111) position, large hBN/Pt(111) areas develop, which alternate with densely bunched steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' At larger vicinal angles the step bunching process remains, and the surface becomes a faceted structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' At the (554) position one observes a rather well ordered structure consisting of (111) terraces and side facets tilted at approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 6◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' At the (223) A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride 8 Vicinal angle a [112] cPt(111) hBN Pt(223) Pt(554) B-steps A-steps 15° 0° 0° 15° 30nm hBN a = -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5° a = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='2° a = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='8° hBN +Eu +Eu 0 40 80 120 0 30 Length (nm) h (nm) 0 0 20 100 200 80nm +Eu 0 0 15 200 400 (111) side facets (111) side facets (111) side facets 40nm Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' STM images of the (Eu)/hBN/c-Pt(111) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Large scale STM images of a hBN monolayer prior and after Eu intercalation collected at three different positions on the curved c-Pt(111) substrate (scanning parameters: I=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='13 nA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' U=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5 V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The line scans were taken close to the top of the images corresponding to the Eu intercalated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' (111) and side facets are indicated by different color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' position one does not get a clear long range order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The latter situation is similar to stepped Ni crystals covered by hBN [56], while the ordered structures are rather close to the Rh case, where hBN growth on stepped surfaces leads to periodically arranged nanofacets [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' One interesting question is whether the hBN forms a continuous, defect- free coat over the hill-and-valley structure underneath, since this requires “bending” of the hBN layer over Pt substrate facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' In general terms, however, one expects an increasing number of defects for an increasing presence of facet/step boundaries at large vicinal angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Eu deposition and intercalation on the vicinal surfaces changes the facet periodicity, size and inclination, as observed in the STM images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' At the A-step type Pt(223) position, the rather disordered hBN/Pt(223) structure transforms into a well ordered array after Eu intercalation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' On B-type steps, however, the Eu intercalation is leading to smaller facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' A statistical analysis of the STM images reveals an increasing average facet distance [(111)+side facet] of 20 facets/µm close to (111), 25 facets/µm at (223), and 65 facets/µm at (554).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This means that at A-type steps the surface rugosity decreases, contrary to B-type steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' On the other hand, XPS spectra (see below) reveal a similar Eu2+/Eu3+ relation on the different crystal positions, with a slightly higher di-valent amount at (111) compared to stepped surface planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Magnetism of the hBN/EuPt2 system in the (111) plane Divalent Eu (4f 7, J = 7/2) has a low-temperature ferromagnetic state in bulk EuPt2 and similar compounds [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' We measured the magnetic properties of our hBN-protected A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 1180 1160 1140 1120 Photon energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 µ+ µ− µ− − µ+ T = 7 K q = 70° 2+ Eu 3+ Eu TEY intesnity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=') (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='9 1180 1160 1140 1120 m H = 6 T 0 XMCD Eu M4,5 2+ Eu 3+ Eu XAS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='15 6 4 2 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='00 30 20 10 0 paramagnetic ferro- magnetic m H/M (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=') 0 2 M (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=') XMCD intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=') Applied field m H (T) 0 2+ Eu mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' curve M4 M5 q = 70° Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Magnetic properties of intercalated Eu below hBN/Pt(111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' (a) X-ray absorption spectrum (total electron yield - TEY) of horizontally (top, XAS) and circularly polarized light with opposite sign and its resulting difference spectrum shown below (XMCD) of 3˚A Eu intercalated below hBN/Pt(111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The XAS spectrum was fitted with several gaussian profiles (black, yellow, green) and a linear background (grey) for each of the two main Eu2+ and Eu3+ contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' (b) Magnetization curve taken at the maximum of the Eu M5 XMCD signal for a variable applied field from +6 T to -6 T (blue) and in the opposite direction (orange), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The sample was oriented with an angle of 70◦ with respect to the applied field at temperature T = 7K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The inset displays the Arrot plot analysis (see text) confirming a ferromagnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The black line is a linear fit to the high field values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' EuPt2 surface alloy with X-ray circular magnetic dichroism (XMCD) at the (111) position in the curved crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The X-ray absorption (XAS) spectrum in part (a) reveals a mixture of Eu2+ and Eu3+ contribution after Eu intercalation below the hBN/Pt(111) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This observation confirms the coexistence of the two Eu valences and it is consistent with the XPS results, namely di-valent Eu atoms at the EuPt2 surface and tri-valent Eu atoms diffused into the Pt bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The XMCD signal results from the difference of the absorption spectra of left and right circularly polarized light and is shown in the bottom of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' At the applied field A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride 10 of 6 T, it shows the typical lineshape of pure di-valent Eu [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This is expected since, as stated before, Eu3+ has a 4f 6 configuration with S = L = 3 and J = 0, hence the Eu3+ signal should not contribute significantly to the anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Ferromagnetism can be probed by measuring the Eu XMCD signal while varying the applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The XMCD signal is proportional to the magnetization M in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The resulting magnetization curve is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' It reveals a pronounced “S” shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The Arrot plot analysis [60] shown in the inset confirms the ferromagnetic state of the EuPt2 surface alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' In such analysis, the square of the magnetization M 2 is plotted against the applied field divided by the magnetization µ0H/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' A linear fit of the high field (high µ0H/M) values indicates ferromagnetic state if the line hits the ordinate at a positive M 2 value, and paramagnetism if the ordinate crossing is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Therefore, the Arrot plot reveals ferromagnetic behavior at the intercalated hBN/Eu- Pt(111) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Nevertheless, the magnetization curve shows that both, remanence and coercive field, are too small to be detectable at the measurement temperature of 7K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' For the out-of-plane geometry with the sample normal to the magnetic field and light incidence, the magnetization curves are slightly more rectangular at small fields pointing to an out-of-plane easy axis (see supplementary material for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Exposure of the hBN/Eu/Pt system to air Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 6(a) displays the complete evolution of the XPS spectra during the intercalation of 4-˚A-Eu in the hBN/Pt(111) interface and after exposure to air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Prior to Eu intercalation, we obtain the characteristic shape and positions in the B 1s and N 1s core levels for the weakly-coupled hBN/Pt(111) system [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' After Eu intercalation B 1s and N 1s core levels notably change their shape and energy, reflecting the fact that the hBN contact interface is now different (EuPt2/hBN), with a stronger hBN-EuPt2 interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' With respect to the Eu core level, as mentioned above, it proves that Eu atoms are present in two configurations, di-valent Eu2+ for the 2D EuPt2 alloy and tri-valent Eu3+ for Eu atoms incorporated into the Pt bulk below the EuPt2 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' After sample exposition to ambient conditions (6 hours, room temperature, 80% humidity) and a vacuum annealing to 770K to remove impurities from the air exposure, the ratio of Eu2+/Eu3+ drops to one third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The O 1s core level is detected at the higher binding energy side of the Pt 4p3/2 spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' A detailed view (inset) indicates a double-peak with energy positions of 530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5 eV and 532 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The former is in good agreement with the binding energy in the tri-valent Eu2O3 oxide [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The 532 eV emisssion is interpreted as due to hydroxide -OH, typical from Eu samples exposed to air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' On the other hand, B 1s and N 1s core levels partially recover their shape and energy prior to Eu exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This indicates that the hBN interaction with partially oxidized EuPt2 patches underneath is weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' On the other hand, oxidation of the hBN layer is not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The hBN protection for the intercalated EuPt2 alloy in vicinal Pt substrates is examined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 6(b), in direct comparison with the Pt(111) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Here we show the Eu 3d, Pt 4p3/2 and O 1s core levels at the Pt(332) and Pt(223) surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' At the Pt(332) A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='6 196 194 192 190 188 B 1s 16 14 12 10 8 PE intensity (counts/sec) 402 400 398 396 N 1s 18 16 14 12 10 8 6 580 560 540 520 500 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='6 538 530 hBN +4 Å Eu, 870K +air 12 10 8 6 4 1180 1160 1140 1120 1100 O 1s Pt 4p3/2 Eu 3d N Auger 2+ Eu 3+ Eu 14 12 10 8 6 580 560 540 520 500 11 10 9 8 538 530 O 1s Pt 4p3/2 14 12 10 8 6 1180 1160 1140 1120 1100 Eu 3d N Auger 2+ Eu 3+ Eu +air (111) (332) (223) Eu/hBN E-E (eV) F (a) (b) 2+ Eu 3+ Eu 2+ Eu 3+ Eu Pt(111) curved Pt Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' XPS analysis of Eu intercalation below hBN on the curved Pt crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' (a) N 1s, B 1s, Eu 3d, O 1s, and Pt 4p3/2 X-ray photoemission spectra taken at hν=1486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='6 eV (Al Kα) for the Eu 4 ˚A preparation before and after exposing to ambient conditions at Pt(111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' (b) Comparative Eu 3d, O 1s, and Pt 4p3/2 core levels at three positions of the curved Pt substrate, at Pt(332), Pt(111) and Pt(335) positions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' plane, no di-valent signal reamins in the Eu 3d spectrum after exposure to air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' At the Pt(223) position the Eu2+/Eu3+ is strongly reduced, but the Eu2+ peak is still visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Again, the O 1s spectrum suggests that the strong reduction of the number of di-valent Eu atoms is due to the formation of tri-valent Eu oxides and hydroxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The overall O 1s intensity increases as the Eu2+ peak decreases at the stepped surfaces, although a more detailed peak analysis indicates that the intensity of the Eu-OH peak at 532 eV is similar in all three cases, and it is the 530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5 eV peak from Eu2O3 the one that scales reciprocally with the Eu2+/Eu3+ ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The complete oxidation of the B-type (332) surface correlates with the high facet density observed in the STM analysis of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 4, since this allows a higher number of hBN bending or breaks at facet borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The rather small divalent signal that remains in Pt(223) may reflect the presence of larger (111) and side facets where the intercalated EuPt2 alloy remains better protected at ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Finally, we analyze the impact of the air exposure on the hBN protecting layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' For this purpose angle-resolved photoemission spectroscopy (ARPES) is particularly appropriate, since it is highly sensitive to the hBN valence band, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The photoemission intensity map of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 7(a) corresponds to the hBN/Pt(111) interface (centre of the curved sample), and has been measured using the He II excitation energy (hν = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='8eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The strongly dispersing, intense features are the hBN π bands, with A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride 12 12 10 8 6 4 2 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 12 10 8 6 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 hBN +1 Å Eu +5min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' air +annealing E-E (eV) F 1 |k | (Å ) || (a) (b) (d) (c) G K K G K K hBN/EuPt2 hBN/Pt hBN/EuPt2 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Angle resolved Photoemission analysis of the hBN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' He IIa (hν = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='8eV) photoemission intensity maps along ¯Γ ¯K direction of the hBN band structure for (a) hBN/Pt(111), (b) after intercalation of 1 ˚A of Eu at T = 770K, (c) exposure of that sample to air, and (d) after another annealing to 770K to desorb air contamination, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The most intense features correspond to the π-band of hBN at the indicated interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' minimum at the ¯Γ point of the Brillouin zone and maximum at ¯K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The emission between approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 2 and 8 eV binding energy entirely belongs to hBN, while the Pt valence band features appear closer to the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' After intercalation of a very small amount of Eu (1 ˚A), the main π band appears unaltered, although a replica of this band emerges below, at 2 eV higher binding energy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 7(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The unaltered band corresponds to pure Pt areas without Eu while the shifted band arises in areas where the Eu intercalates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The band shift is explained by the enhanced interaction of the hBN with the EuPt2 substrate, with a net electron transfer from Eu atoms to the hBN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' After air exposure all bands get quite blurry, due to surface contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The dominating hBN π band is still visible, slightly shifted to higher binding energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Interestingly, both the pure hBN/Pt and the Eu-intercalated bands can be recovered by annealing the sample again to 770K, which removes adsorbates from the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This indicates that the oxygen intercalation does not affect (oxidize) the hBN layer, which remains intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The photoemission intensity mapping still includes the higher binding energy replica, indicating that strongly interacting hBN/EuPt2 patches remain, despite the partial oxidation of the EuPt2 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Conclusions We have investigated the structural, magnetic and electronic properties of Eu after intercalation between a Pt substrate and a hBN monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' We used a Pt sample curved around the (111) direction in order to additionally assess the role of substrate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Our LEED analysis of the (111) interface shows a ∼( √ 3 × √ 3)R30◦ pattern, revealing the presence of the EuPt2 surface alloy under the hBN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' We find that Eu A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride 13 atoms in this EuPt2 layer are divalent, while Eu atoms that have diffused further inside the Pt bulk during the intercalation process are trivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Interestingly, the Eu2+/Eu3+ ratio is not affected by the presence of steps at the Pt substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' XMCD magnetization curves on the di-valent Eu atom reveal a ferromagnetic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Air exposure of the sample leads to a partial protection of the magnetic, divalent Eu atoms at the (111) plane, while at vicinal surfaces the protecting role of the hBN layer is less efficient, as reflected in the larger attenuation of the divalent Eu state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Such incomplete protection of vicinal planes may be related to a larger number of defects and domain boundaries in a more discontinuous hBN layer, since this covers a much rougher hill-and-valley faceted structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This facilitates oxygen diffusion, intercalation and the EuPt2 alloy oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' In contrast, the hBN layer itself remains intact upon both Eu intercalation and air exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Supplementary material Supplementary material contains information on electron spectroscopy analysis for Eu intercalation at insufficient substrate temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Furthermore, the magnetic easy axis direction is investigated by XMCD technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Acknowledgments We acknowledge financial support from grants PID2020-116093RB-C44 funded by the Spanish MCIN/AEI/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='13039/501100011033 and the Basque Government (Grant IT- 1591-22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' We acknowledge the European Synchrotron Radiation Facility for provision of beam time on ID32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' ESRF access was provided through proposal MA-5454 [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Part of the research leading to the result has been supported by the project CALIPSOplus under Grant Agreement 730872 from the EU Framework Programme for Research and Innovation HORIZON 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' appreciates the support of Japan Society for the Promotion of Science (JSPS) Overseas Research Fellowships and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' and F.' metadata={'source': 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Wang K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Du S and Xiao X 2020 Two-Dimensional Rare Earth-Gold Intermetallic Compounds on Au(111) by Surface Alloying The Journal of Physical Chemistry Letters 11(0) 4107–4112 A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride 17 [54] Ormaza M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Fern´andez L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Ilyn M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Maga˜na A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Xu B,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Hedo M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Nakama T and ¨Onuki Y 2016 Magnetic and Fermi Surface Properties of Ferromagnets EuPd2 and EuPt2 Journal of the Physical Society of Japan 85(8) 084705 [59] Blanco-Rey M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Castrillo-Bodero R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Ali K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Gargiani P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Bertran F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Sheverdyaeva P M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Ortega J E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Fernandez L and Schiller F 2022 Effect of the valence state on the band magnetocrystalline anisotropy in two-dimensional rare-earth/noble-metal compounds Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Research 4 013237 [60] Arrott A 1957 Criterion for ferromagnetism from observations of magnetic isotherms Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 108 1394–1396 [61] Mercier F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Alliot C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Bion L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Thromat N and Toulhoat P 2006 XPS study of Eu(III) coordination compounds: Core levels binding energies in solid mixed-oxo-compounds EumXxOy Journal of Electron Spectroscopy and Related Phenomena 150(1) 21–26 [62] Baltrus J P and Keller M J 2019 Rare earth oxides Eu2O3 and Nd2O3 analyzed by XPS Surface Science Spectra 26(1) 014001 [63] Mohammed Idris Bakhit A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Schiller F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Fernandez Gomez-Recuero L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Castrillo Bodero R and Hasegawa Y 2022 Magnetic properties in the single molecular magnet TbPc2 insulating ferromagnet EuO2 system https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='15151/ESRF-ES-886076657 Supplementary material of “A ferromagnetic Eu-Pt surface compound grown below hexagonal boron nitride” Alaa Mohammed Idris Bakhit1,2, Khadiza Ali3, 4, Anna A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Makarova5, Igor P´ıˇs6, Federica Bondino6, Roberto Sant7, Saroj P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Dash4, Rodrigo Castrillo1, Yuri Hasegawa1,8, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Enrique Ortega1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Laura Fernandez1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' and Frederik Schiller1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='3 1 Centro de F´ısica de Materiales CSIC-UPV/EHU-Materials Physics Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' E-20018 San Sebasti´an,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Spain 2 Departamento de Pol´ımeros y Materiales Avanzados: F´ısica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Qu´ımica y Tecnolog´ıa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Universidad del Pa´ıs Vasco UPV/EHU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 20018 San Sebasti´an,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Spain 3 Donostia International Physics Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' E-20018 Donostia-San Sebasti´an,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Spain 4 Chalmers University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' G¨oteborg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Chalmersplatsen 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 412 96 G¨oteborg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Sweden 5 Physikalische Chemie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Institut f¨ur Chemie und Biochemie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Freie Universit¨at Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Arnimallee 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 14195 Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Germany 6 IOM-CNR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Strada Statale 14 Km 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5, I-34149 Trieste, Italy 7 ESRF, The European Synchrotron, 71 Avenue des Martyrs, CS40220, 38043 Grenoble Cedex 9, France 8 Department of Physical Sciences, Ritsumeikan University, Kusatsu, 525-8577, Japan 9 Departamento de F´ısica Aplicada I, Universidad del Pa´ıs Vasco UPV/EHU, E-20018 San Sebasti´an, Spain E-mail: frederikmichael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='schiller@ehu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='es The supplementary information file contains: Supplementary Figures S1-S4 Supplementary information and data are provided to further illustrate the intercalation process of Eu below the hBN/Pt substrate as well as magnetic easy axis determination of the Eu-Pt alloy below hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='11837v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='mtrl-sci] 27 Jan 2023 Supplementary material S2 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Spectroscopy analysis of insufficient temperature for Eu intercalation Eu does not completely intercalate below a hBN layer on Pt if the substrate temperature is not sufficient high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Here we will show Near-edge X-ray absorption fine structure (NEXAFS) and X-ray photoelectron spectroscopy (XPS) data for such preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Data were taken at ID32 and BACH beamlines of ESRF and Elettra synchrotrons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' X-ray absorption spectroscopy (normal to the field and light incidence) for a approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 1 ML thick Eu film deposited at a T = 610K hot hBN/Pt(111) substrate and successive exposure to air is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The air exposure was carried out at with the sample at room temperature during five minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' After air exposure a strong transformation of the spectral shape towards tri-valent Eu configuration took place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Nevertheless, even after the air exposure experiment a small di-valent part is still present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' In order to remove remaining air contamination, the sample was post-annealed in UHV to T = 470K during 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' S2 represents X-ray photoemission data from BACH beamline taken at a photon energy hν = 272eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The substrate temperature during Eu deposition for the data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' S2 was T = 470K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The data set includes measurements of hBN/Pt(111) and two successive Eu intercalations, each for 10 min at a very low rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The total Eu thickness was calculated from the Pt 4f intensity loss taking into account the mean free path of the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The latter was extracted from the universal curve of the electron mean free path [1] and resulted in Eu thickness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='7 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5˚A for 10 and 20 min Eu deposition, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' S2 one observes the spin orbit split 4f7/2 and 4f5/2 components of the Pt 4f core level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' These are further split in the surface (S) and bulk (B) emissions indicated in the Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' It is interesting to note that the surface emission can be still observed for hBN/Pt(111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This is possible due to the weak interaction of hBN and the Pt, specially at the (111) face that hosts a hBN Moire 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='0 TEY intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=') 1190 1180 1170 1160 1150 1140 1130 1120 Photon energy hn (eV) T = 300 K q = 0° Eu M4,5 Eu/hBN/Pt(111) +Air exposure 2+ Eu 3+ Eu M5 M4 Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' X-ray absorption spectra at the Eu M4,5 absorption edge for Eu deposition on hBN/Pt(111) at T = 610K and successive air exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Supplementary material S3 PE intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=') 200 150 100 50 0 E-E (eV) F hBN 10 min Eu @ 470K 20 min Eu @ 470K hn = 272 eV B 1s Pt 4f 194 192 190 188 A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='93 AhBN ® 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='35 Å Eu A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='83 AhBN ® 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='9 Å Eu AhBN B 1s 76 74 72 70 pure Pt(111) hn = 190eV A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='95 AhBN ® 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='7 Å Eu A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='87 AhBN ® 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5 Å Eu AhBN/Pt(111) Pt 4f S B S B 4f7/2 4f5/2 Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' XPS analysis for sub-monolayer Eu deposition on hBN/Pt(111) at T = 470K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' XPS survey, B 1s and Pt 4f emissions of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='7 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5˚A Eu coverage on hBN/Pt(111), respectively, taken for hν = 272eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' In the case of the Pt 4f also the clean Pt emission is included to better observe the surface and bulk contributions (hν = 190eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' After low binding energy normalization, the areas A below the emissions were extracted and from them the overlayer amount was calculated by taking into account the electron mean free path of the electrons from the standard curve [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' From this analysis we observe that not all Eu is intercalated below the hBN layer but rather stay on top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' After Eu deposition the surface component shrinks much more than the bulk component, revealing that partial Eu intercalation occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' We again calculate the Eu overlayer thickness but now from the B 1s core level that give us the amount of Eu on top of the hBN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The thickness that results from the area diminution is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='35 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='9˚A, respectively, for the two evaporations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This means that during the first deposition 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='35˚A intercalated and after the second deposition 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='6˚A out of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5˚A Eu intercalated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' We also observe the doping effect of Eu towards the hBN layer that shifts the B 1s core level to higher binding energies in a similar way as for complete intercalation, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 6(a) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The analysis of the valency has been carried out with resonant photoemission applying photon energies around the 4d→4f absorption edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The results can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' At off-resonant photoemission (hν = 132eV) the Eu 4f emission is strongly suppressed, therefore the valence band is dominated by Pt 5d valence band emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Nevertheless, there is already some Eu 4f emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The 4f emissions are much better observed for the resonant photon energies hν = 140eV and 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5eV corresponding to Eu2+ and Eu3+ resonances, respectively [2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The di-valent emission is located at approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='1eV binding energy, the tri-valent multiplet is observed between 4 and 12eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Due to the unknown cross-section effects around the resonances an exact Eu2+/Eu3+ ratio cannot be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The tri-valent contribution arises from a part Supplementary material S4 300 250 200 150 100 50 0 30 25 20 15 10 5 0 on-resonant 2+ Eu 200 150 100 50 0 30 25 20 15 10 5 0 60 40 20 0 30 25 20 15 10 5 0 off-resonant 2+ Eus EuO 3+ Eu 4f Eu 5p hn = 132eV hn = 140eV hn = 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5eV on-resonant 3+ Eu 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5Å Eu @ 470K +air exposure 3 PE intensity (10 counts) E-E (eV) F Pt 5d+ Eu 4f Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Resonant Photoemission at the Eu 4d→4f absorption edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Normal emission photoemission spectra of the valence band region of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5˚A Eu deposited onto a 470K hot hBN/Pt(111) surface for three photon energies corresponding to off, on-resonant Eu2+, and on-resonant Eu3+ at hν = 132, 140, and 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='5eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Shown are spectra prior and after exposure of the sample to ambient conditions (5 min, room temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Supplementary material S5 of the intercalated Eu atoms that diffuse further inside the Pt bulk, but can also arise from oxidation of Eu to tri-valent Eu in Eu2O3 of the Eu atoms atop of the hBN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The latter oxidation is very difficult to avoid due to the strong reactivity of Eu even under very good ultra-high vacuum conditions [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' After air exposure, the situation changes drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' There is still a small di-valent signal, but now at a binding energy of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='3eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This Eu2+ emission cannot arise from di-valent Eu surrounded by a metallic environment whose binding energy is always lower than 2eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' But an emission at such 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='3eV arises from divalent Eu in EuO at surfaces [6, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' This oxide is usually unstable under ambient conditions but seems to be able to form and remain stable under used experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' EuO is created either on the surface or at the interface under hBN (and stabilized thanks to the interface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Further investigations would be necessary to distinguish both possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' In any case, we do not observe a possible divalent EuPt2 structure below the hBN after the air exposure process for Eu deposition at 470K substrate temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Magnetic anisotropy determination The magnetic anisotropy of the intercalated Eu can be determined from the two different geometries applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The magnetization curves for the sample perpendicular and at an angle of 70◦ with respect to the applied field is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The close-up at small fields reveal a more rectangular shape of the out-of-plane geometry magnetization loop pointing to this direction as the easy axis of magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The XMCD values close to zero-field has been taken from individual complete XMCD measurements to avoid the typical zero-field artifacts of XMCD measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='15 XMCD signal (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=') 6 4 2 0 2 4 6 Applied field m H (T) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content='05 1 0 1 θ = 0° θ = 70° θ = 0° θ = 70° Applied field m H (T) 0 XMCD (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=') 2+ Eu mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' curve Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Eu magnetization curves at different geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Eu magnetization curves normal and at θ = 70◦ with respect to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' The inset shows a closeup at small fields to distinguish better the two curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' One clearly observes the more rectangular like shape for the out-of-plane geometry revealing an out-of-plane easy axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Supplementary material S6 [1] Dench W A and Seah M P 1979 Quantitative Electron Spectroscopy of Surfaces: A Standard Data Base for Electron Inelastic Mean Free Path in Solids Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Interf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' 1 1 [2] Schneider W D, Laubschat C, Kalkowski G, Haase J and Puschmann A 1983 Surface effects in eu intermetallics: A resonant photoemission study Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' B 28 2017–2022 [3] Santana J A C, Liu P, Wang X, Tang J, McHale S R, Wooten D, McClory J W, Petrosky J C, Wu J, Palai R, Losovjy Y B and Dowben P A 2012 The local metallicity of gadolinium doped compound semiconductors Journal of Physics: Condensed Matter 24(44) 445801 [4] Banik S, Bendounan A, Thamizhavel A, Arya A, Risterucci P, Sirotti F, Sinha A K, Dhar S K and Deb S K 2012 Electronic structure of eucu2ge2 studied by resonant photoemission and x-ray absorption spectroscopy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' B 86 085134 [5] Schumacher S, Huttmann F, Petrovi´c M, Witt C, F¨orster D F, Vo-Van C, Coraux J, Mart´ınez- Galera A J, Sessi V, Vergara I, R¨uckamp R, Gr¨uninger M, Schleheck N, Meyer zu Heringdorf F, Ohresser P, Kralj M, Wehling T O and Michely T 2014 Europium underneath graphene on Ir(111): Intercalation mechanism, magnetism, and band structure Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' B 90 235437 [6] Caspers C, M¨uller M, Gray A X, Kaiser A M, Gloskovskii A, Fadley C S, Drube W and Schneider C M 2011 Chemical stability of the magnetic oxide euo directly on silicon observed by hard x-ray photoemission spectroscopy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} +page_content=' B 84 205217' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFKT4oBgHgl3EQfhS5q/content/2301.11837v1.pdf'} diff --git a/QNFOT4oBgHgl3EQf5TRc/content/tmp_files/load_file.txt b/QNFOT4oBgHgl3EQf5TRc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d1d2b32885417c0fc1141db603940aa5e182cc0 --- /dev/null +++ b/QNFOT4oBgHgl3EQf5TRc/content/tmp_files/load_file.txt @@ -0,0 +1,322 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf,len=321 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='12953v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='RA] 30 Sep 2022 The ω-Lie algebra defined by the commutator of an ω-left-symmetric algebra is not perfect Zhiqi Chen School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' E-mail: chenzhiqi@nankai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='cn Junna Ni Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Department of Mathematics, South China Normal University, Guangzhou 510631, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Email: nijunna@126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='com Jianhua Yu Department of Mathematics, South China Normal University, Guangzhou 510631, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Email: yujianhuscnu@126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='com January 31, 2023 Abstract In this paper, we study admissible ω-left-symmetric algebraic structures on ω-Lie al- gebras over the complex numbers field C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Based on the classification of ω-Lie algebras, we prove that any perfect ω-Lie algebra can’t be the ω-Lie algebra defined by the commutator of an ω-left-symmetric algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 17A30,17B60 Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' ω-Lie algebra, perfect ω-Lie algebra, ω-left-symmetric alge- bra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 1 Introduction A vector space L over F is called an ω-Lie algebra if there is a bilinear map [·, ·] : L × L → L and a skew-symmetric bilinear form ω : L × L → F such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' [x, y] = −[y, x], 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' [[x, y], z] + [[y, z], x] + [[z, x], y] = ω(x, y)z + ω(y, z)x + ω(z, x)y, hold for any x, y, z ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Clearly, an ω-Lie algebra L with ω = 0 is a Lie algebra, which is called a trivial ω-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Otherwise, L is called a nontrivial ω-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Similar to Lie algebraic case, an ω-Lie algebra L is called perfect if [L, L] = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' The notation of an ω-Lie algebra is given by Nurowski in [17], and then there are many studies in this field such as [6, 7, 8, 9, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' For an ω-Lie algebra L, a finite dimensional vector space V is called an L-module if there exists a bilinear map L × V → V defined by (x, v) �→ xv such that [x, y]v = x(yv) − y(xv) + ω(x, y)v holds for x, y ∈ L and v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It is well-known that left-symmetric algebras are defined by the module of Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Similarly, ω-left-symmetric algebras are defined ([19]) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 1 Let L be a vector space over F with a bilinear map (x, y) �→ xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' If there is a bilinear map ω : L × L → F such that (xy)z − x(yz) − (yx)z + y(xz) = ω(x, y)z, ∀x, y, z ∈ L, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='1) then L is called an ω-left-symmetric algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Clearly an ω-left-symmetric algebra L with ω = 0 is a left-symmetric algebra, which is called a trivial ω-left-symmetric algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Otherwise, L is called a nontrivial ω-left-symmetric algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Left-symmetric algebras (or pre-Lie algebras, quasi-associative algebras, Vinberg algebras and so on) are first introduced by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Cayley in 1896 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' They appear in many fields in mathematics and mathematical physics, for more details see [1, 2, 3, 5, 11, 12, 13, 14, 16, 18] and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' For an ω-left-symmetric algebra L, define the commutator [x, y] = xy − yx, then L is an ω-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Moreover the left multiplication lx on the ω-left-symmetric algebra L defined by lxy = xy makes L to be an ω-Lie algebra L-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It is an important result that the Lie algebra defined by the commutator of a left-symmetric algebra is not a perfect Lie algebra ([15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' In this paper, we prove this result holds for ω-Lie algebras and ω-left-symmetric algebras, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='1 Let L be an ω-left-symmetric algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then the ω-Lie algebra defined by the commutator [x, y] = xy − yx is not a perfect ω-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Throughout this paper, all vector spaces and algebras are finite dimensional over C unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 2 Perfect ω-Lie algebras By the classification of nontrivial ω-Lie algebras, a nontrival perfect ω-Lie algebra L is one of the following cases: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' dim L = 3, and L is Aα, or B, or Cα for α ̸= 0, −1 in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' There exists a basis {x, y, z} of L such that the nonzero brackets and ω are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Aα : [x, y] = x, [x, z] = x + y, [y, z] = z + αx, ω(y, z) = −1, where α ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' B : [x, y] = y, [x, z] = y + z, [y, z] = x, ω(y, z) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Cα : [x, y] = y, [x, z] = αz, [y, z] = x, ω(y, z) = 1 + α, where 0, −1 ̸= α ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' dim L = 4, and L is G1,α, or H1,α, or � Aα, or �B, or � Cα for α ̸= 0, −1 in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' there exists a basis {x, y, z, e} of L such that the nonzero brackets and ω are G1,α : [e, x] = e + αy, [e, y] = −e + x, [y, z] = z, [x, y] = y, ω(e, x) = α, ω(x, y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' H1,α : [e, x] = e + αy, [e, y] = −e + x + z, [y, z] = z, [x, y] = y, ω(e, x) = α, ω(x, y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' � Aα : [x, y] = x, [x, z] = x + y, [y, z] = z + αx, [e, z] = e, ω(y, z) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' �B : [x, y] = y, [x, z] = y + z, [y, z] = x, [e, x] = −2e, [e, y] = −e, ω(y, z) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' � Cα (α ̸= 0, −1) : [x, y] = y, [x, z] = αz, [y, z] = x, [e, x] = −(1 + α)e, ω(y, z) = 1 + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' dim L ≥ 5, and L = Ch0 ⊕ H1 ⊕ Cx ⊕ Cv, and the nonzero brackets and ω are given as follows: for any h ∈ H1, [x, h0] = −ah0, [v, h] = 1 ah, [v, h0] = h2 + 1 ah0 + x, [x, v] = h1 + av, ω(x, v) = 1, where h1 ∈ Ch0 ⊕ H1, h2 ∈ H1 and a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It is type-P1 in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' dim L ≥ 5, and L = H ⊕ Cx ⊕ Cy ⊕ Ca, and the nonzero brackets and ω are given as follows: for any h ∈ H, [a, h] = h, [x, y] = h3 + a, [x, a] = h1 + b1x + b2y, [y, a] = h2 + c1y, ω(x, y) = 1, where h1, h2, h3 ∈ H, b1 ̸= 0, c1 ̸= 0 and b1 + c1 + 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It is type-P2 in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 3 ω-left-symmetric algebraic structure on perfect ω-Lie alge- bras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' This section is to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' That is, we extend the classical result to ω-Lie algebras: a Lie algebra admitting a left-symmetric algebraic structure is not a perfect Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Assume that L is an ω-left-symmetric algebra such that the ω-Lie algebra defined by the commutator [x, y] = xy − yx is a perfect ω-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Denote it also by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It is enough to discuss the case when L is not a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Thus L is someone in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Define lu : L → L by lu(v) = uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' By the definition of ω-left-symmetric algebra, l[u,v] = [lu, lv] + ω(u, v)id, ∀u, v ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' We will discuss perfect ω-Lie algebras in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' case by case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Case 1: dim L = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' In [17], Nurowski classified nontrivial ω-Lie algebras in dimension 3 over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Based on the classification, the classification of ω-left-symmetric algebras in di- mension 3 is given in [10] as follows: if L is a nontrivial ω-left-symmetric algebra over R in dimension 3, then V has a basis {e1, e2, e3} such that one of the following cases holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' e1e1 = e2e1 = a1e1 + a2e2 + a3e3 = 2e1 − e3e1, e1e2 = e2e2 = (a1 − 1)e1 + (a2 + 1)e2 + a3e3 = 2e3 − e3e2, e1e3 = e2e3 = (2 − a1)e1 + (1 − a2)e2 + (1 − a3)e3 = 2e3 − e3e3, ω(e1, e2) = 0, ω(e2, e3) = 2, ω(e3, e1) = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' e1e1 = 2e1, e1e2 = 2e2, e1e3 = 2e3, e2e1 = e3e1 = e2 + e3, e2e2 = e3e2 = a1e1 + a2e2 + a3e3, e2e3 = e3e3 = (a1 + 1)e1 + a2e2 + a3e3, ω(e1, e2) = 0, ω(e2, e3) = 2, ω(e3, e1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then by the above classification of ω-left-symmetric algebras in dimension 3 over R, we only need to discuss L = Aα and L = Cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' For the case L = Aα, the nonzero brackets and ω are given as follows: [x, y] = x, [x, z] = x + y, [y, z] = z + αx, ω(y, z) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then we have: lx = [lx, ly], lx + ly = [lx, lz], lz + αlx = [ly, lz] − id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It follows that lx + ly = [lx, lz] = [[lx, ly], lz] = [[lx, lz], ly] + [lx, [ly, lz]] = [lx, ly] + [lx, lz] = 2lx + ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' That is, lx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It follows that ly = 0 and lz = −id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then x = [x, y] = lxy − lyx = 0, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' For the case L = Cα, the nonzero brackets and ω are given as follows: [x, y] = y, [x, z] = αz, [y, z] = x, ω(y, z) = 1 + α, 3 where α ̸= 0, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then we have: ly = [lx, ly], αlz = [lx, lz], lx = [ly, lz] + (1 + α)id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It follows that lx = [ly, lz] + (1 + α)id = [[lx, ly], lz] + (1 + α)id = [[lx, lz], ly] + [lx, [ly, lz]] + (1 + α)id = α[lz, ly] + (1 + α)id = −αlx + (1 + α)2id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' That is, lx = (1+α)id since α ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It follows that ly = lz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then x = [y, z] = lyz−lzy = 0, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Case 2: dim L = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' we will discuss case by case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' The first case is L = G1,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then the nonzero brackets and ω are given as follows: [e, x] = e + αy, [e, y] = −e + x, [y, z] = z, [x, y] = y, ω(e, x) = α, ω(x, y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Let e′ = e + αy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then for {e′, x, y, z}, we have: [e′, x] = e′ − αy, [e′, y] = −e′ + x + αy, [e′, z] = αz, [y, z] = z, [x, y] = y, ω(x, y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then we have: ly = [lx, ly] + id, [lx, lz] = 0, lz = [ly, lz], le′ − αly = [le′, lx], −le′ + lx + αly = [le′, ly], αlz = [le′, lz].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Clearly −le′ + lx + αly = [le′, ly] means le′ = lx + αly − [le′, ly].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then le′ − αly = [lx + aly − [le′, ly], lx] = α[ly, lx] − [[le′, ly], lx] It is easy to see that [[le′, ly], lx] = [[le′, lx], ly] + [le′, [ly, lx]] = [le′, ly] − [le′, ly] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' That is, le′ − aly = α[ly, lx] = α(id − ly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It means le′ = αid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then we have αly = aid, αlz = 0, lx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then e′ − αy = [e′, x] = e′x − xe′ = le′x − lxe′ = αx, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' The second case is L = H1,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' For this case, the nonzero brackets and ω are [x, y] = y, [y, z] = z, [e, y] = x + z − e, [e, x] = e + αy, ω(x, y) = 1, ω(e, x) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 4 It follows that ly = [lx, ly] + id, lz = [ly, lz], [lx, lz] = 0, lx + lz − le = [le, ly], le + αly = [le, lx] + αid, [le, lz] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Clearly, 0 = [lx, lz] = [lx, [ly, lz]] = [[lx, ly], lz] + [ly, [lx, lz]] = [ly, lz] = lz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Clearly, le = [le, lx] + αid − αly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then we have lx − le = [le, ly] = [[le, lx] + αid − αly, ly] = [[le, lx], ly] = [[le, ly], lx] + [le, [lx, ly]] = −[le, lx] + [le, ly].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It gives le = α(id − ly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then lx = le + [le, ly] = le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Furthermore, ly − id = [lx, ly] = [le, ly] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It follows that ly = id, lx = le = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then y = [x, y] = lxy − lyx = −x, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' The third case is L = � Aα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' The nonzero brackets and ω are given [x, y] = x, [x, z] = x + y, [y, z] = z + αx, [e, z] = e, ω(y, z) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then we have lx = [lx, ly], lx + ly = [lx, lz], lz + αlx = [ly, lz] − id, le = [le, lz], [le, lx] = 0, [le, ly] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Clearly lx = [lx, lz] − ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then lx = [lx, ly] = [[lx, lz] − ly, ly] = [[lx, lz], ly] = [[lx, ly], lz] + [lx, [lz, ly]] = [lx, lz] + [lx, −lz − αlx − id] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then ly = 0 by lx + ly = [lx, lz].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' But x = [x, y] = xy − yx = lx(y) − ly(x) = 0, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' The fourth case is L = �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' The nonzero brackets and ω are [x, y] = y, [x, z] = y + z, [y, z] = x, [e, x] = −2e, [e, y] = −e, ω(y, z) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then we have ly = [lx, ly], ly + lz = [lx, lz], lx = [ly, lz] + 2id, −2le = [le, lx], −le = [le, ly], [le, lz] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Since le = [ly, le], we have −2le = [le, lx] = [[ly, le], lx] = [[ly, lx], le] + [ly, [le, lx]] = [−ly, le] + [ly, −2le] = −3[ly, le] = −3le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 5 That is le = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Since ly = [lx, ly], we have lx = [ly, lz] + 2id = [[lx, ly], lz] + 2id = [[lx, lz], ly] + [lx, [ly, lz]] + 2id = [ly + lz, ly] + [lx, lx − 2id] + 2id = 4id − lx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It means lx = 2id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then ly = lz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' But y = [x, y] = lxy − lyx = 2y, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' The fifth case is L = � Cα (α ̸= 0, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' The brackets and ω are given as follows: [x, y] = y, [x, z] = αz, [y, z] = x, [e, x] = −(1 + α)e, ω(y, z) = 1 + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then we have ly = [lx, ly], αlz = [lx, lz], lx = [ly, lz] + (1 + α)id (α + 1)le = [lx, le], [le, ly] = 0, [le, lz] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Furthermore we have lx = [ly, lz] + (1 + α)id = [[lx, ly], lz] + (1 + α)id = [[lx, lz], ly] + [lx, [ly, lz]] + (1 + α)id = α[lz, ly] + (1 + α)id = −αlx + (1 + α)2id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' That is lx = (1 + α)id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then ly = 0 by ly = [lx, ly].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' But y = [x, y] = lxy − lyx = (1 + α)y, which is impossible since α ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Case 3: P1-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Let x′ = x + h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then [v, h0] = 1 ah0 + x′, [x′, h0] = −ah0 and [x′, v] = h′ 1 + av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Here h′ 1 = h1 − 1 ah2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then we have 1 alh0 + lx′ = [lv, lh0], lh′ 1 + alv + id = [lx′, lv], −alh0 = [lx′, lh0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then we have [lv, −alh0] = [lv, [lx′, lh0]] = [[lv, lx′], lh0] + [lx′, [lv, lh0]] = [−lh′ 1 − alv − id, lh0] + [lx′, 1 alh0 + lx′] = −a[lv, lh0] + 1 a[lx′, lh0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It follows that 1 a[lx′, lh0] = −lh′ 0 = 0, and then lx′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then −ah0 = [x′, h0] = lx′(h0) − lh0(x′) = 0, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Case 4: P2-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' For this case, we have [lx, lh] = 0, [ly, lh] = 0, lh = [la, lh], lh1 + b1lx + b2ly = [lx, la], lh2 + c1ly = [ly, la], lh3 + la = [lx, ly] + id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' By lh2 + c1ly = [ly, la] and [lx, lh] = 0, we have [lx, c1ly] = [lx, [ly, la] − lh2] = [lx, [ly, la]] − [lx, lh2] = [lx, [ly, la]] − [lx, lh2] = [[lx, ly], la] + [ly, [lx, la]] Then by lh3 + la = [lx, ly] + id and lh1 + b1lx + b2ly = [lx, la], we have [lx, c1ly] = [lh3 + la − id, la] + [ly, lh1 + b1lx + b2ly] = −lh3 − b1[lx, ly].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 6 Since c1 + b1 + 1 = 0, we have that [lx, ly] = lh3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' By lh3 + la = [lx, ly] + id, la = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' It follows that lh = ly = lx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Then h3 + a = [x, y] = xy − yx = lxy − lyx = 0, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' In summary, we finish Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Acknowledgements Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Chen was partially supported by NNSF of China (11931009 and 12131012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Bakalov and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Kac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Field algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' (2003): 123–159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Bordemann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Generalized Lax pairs, the modified classical Yang-Baxter equation, and affine geometry of Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 135(1990): 201–216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Burde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Left-symmetric algebras, or pre-Lie algebras in geometry and physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 4(2006): 323–357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Cayley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' On the theory of analytic forms called trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Collected Mathematical Papers of Arthur Cayley, Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Press, 3(1890): 242–246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Chapoton and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Livernet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Pre-Lie algebras and the rooted trees operad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' (2001): 395–408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Liu and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Classification of three dimensional complex ω-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 71(2014): 97–108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' [7] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Chen and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Simple ω-Lie algebras and 4-dimensional ω-Lie algebras over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Malays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content='40(3)(2017): 1377–1390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Zhang and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Zhuang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Derivations, automorphisms, and rep- resentations of complex ω-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' 46(2)(2017): 708–723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' [9] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Ni and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' The classification of ω-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' Preprint, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} +page_content=' [10] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFOT4oBgHgl3EQf5TRc/content/2301.12953v1.pdf'} 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Oppermann1, S Mhatre2, S Gräfe2,3, M Lein1 +1 Leibniz University Hannover, Institute of Theoretical Physics, Appelstraße 2, 30167 +Hannover, Germany +2 Institute of Physical Chemistry, Friedrich Schiller University Jena, Helmholtzweg 4, +07743 Jena, Germany +3 Fraunhofer Institute of Applied Optics and Precision Engineering, +Albert-Einstein-Str. 7, 07745 Jena, Germany +E-mail: lein@itp.uni-hannover.de +Abstract. +We study the effect of the nuclear-mass ratio in a diatomic molecular +ion on the dissociation dynamics in strong infrared laser pulses. A molecular ion is +a charged system, in which the dipole moment depends on the reference point and +therefore on the position of the nuclear center of mass, so that the laser-induced +dynamics is expected to depend on the mass asymmetry. Whereas usually both the +reduced mass and the mass ratio are varied when different isotopologues are compared, +we fix the reduced mass and artificially vary the mass ratio in a model system. This +allows us to separate effects related to changes in the resonance frequency, which is +determined by the reduced mass, from those that arise due to the mass asymmetry. +Numerical solutions of the time-dependent Schrödinger equation are compared with +classical trajectory simulations. +We find that at a certain mass ratio, vibrational +excitation is strongly suppressed, which decreases the dissociation probability by many +orders of magnitude. +Keywords: strong laser fields, helium hydride molecular ion, laser-induced dissociation, +time-dependent Schrödinger equation + +Mass-ratio dependent strong-field dissociation of helium hydride isotopologues +2 +1. Introduction +Molecules under the influence of an external field can undergo various types of excitation +and, in the case of a strong laser field, can be ionized or dissociated [1, 2, 3, 4, 5]. For +long wavelengths of the laser field, a large number of photons is needed to overcome +the electronic excitation energies and hence, electronic transitions become unlikely. The +direct excitation of atomic motion within the electronic ground state, on the other hand, +requires (within the electric-dipole approximation) the presence of a non-zero permanent +electric dipole or at least a change of the dipole with the geometry of the molecule. +For this reason, direct vibrational excitations are dipole-forbidden in homonuclear +diatomic molecules due to their strictly vanishing electric dipole. Heteronuclear diatomic +molecules have a dipole that couples directly to the applied field. +The permanent +dipole moment and the nuclear masses are the relevant parameters that determine the +quantitative amount of vibrational excitation and dissociation. For neutral diatomic +molecules, we are used to the idea that the permanent dipole at a specified internuclear +vector has a well-defined value independent of the nuclear masses. +For a molecular +ion, however, we must take into account the fundamental statement that the dipole +moment of a charged system depends on the choice of reference point. According to +the separation into center-of-mass motion and relative motion, the relevant dipoles for +vibrational excitation (i.e., excitation of the relative motion) must be calculated with +the center of nuclear mass as the reference point. Thus, the dipole moments of molecular +ions depend on the nuclear masses. In a diatomic system, the reduced nuclear mass is, +besides the dipole, the other crucial parameter that determines the nuclear dynamics. +The reduced mass determines the vibrational energy levels and therefore the values of +the resonant transition frequencies. Here it is important to note that the dipole moment +depends on the nuclear masses even when the reduced mass is kept constant. +Isotope effects in photodissociation processes have always been a matter of interest, +and many different molecular species have been studied, ranging from simple diatomic +molecules [6, 7] to polyatomic organic molecules [8, 9]. Nevertheless, the effect of the +mass-dependent electric dipoles has not been isolated since one requires a charged +system, and furthermore, the separation from changes in the reduced mass is not +straightforward. +It would be interesting to observe the isotopologue dependence of +laser-induced dissociation of a molecular ion at fixed reduced mass. Unfortunately, there +are few realistic target systems for such a purpose. In the present work, we consider +artificial isotopologues of the HeH+ molecular ion, which can be considered the simplest +heteronuclear molecule. HeH+ can be prepared in the laboratory [10] and it has recently +been observed in interstellar space [11]. It has already served as an asymmetric polar +benchmark system in a number of studies [12, 13, 14, 15]. Its isotopologues, e. g. 4HeH+, +4HeD+, 3HeH+, etc. possess the same electronic configuration but they differ in both +total and reduced nuclear mass. Most of the isotopologues that could in principle be +constructed from the real isotopes of He and H differ in reduced mass, but there are two +examples, 3HeT+ and 6HeD+, with (approximately) the same reduced mass—albeit not + +Mass-ratio dependent strong-field dissociation of helium hydride isotopologues +3 +Table 1. Properties of selected isotopologues of HeH+. +isotopologue +total mass +reduced mass +mass ratio r = mH/M +4HeH+ +5mn +0.8mn +0.2 +4HeD+ +6mn +1.33mn +0.33 +3HeT+ +6mn +1.5mn +0.5 +6HeD+ +8mn +0.25 +easily experimentally available. (6He has a half-life time of 0.8 s [16].) In Table 1, we +show the properties of selected isotopologues of HeH+, where, for simplicity, protons and +neutrons are idealized as having equal masses mn = 1837 a.u. and the binding energy +(mass defect) of the nuclei is neglected. The reduced mass is given by µ = mHmHe/M +and we define the mass ratio r as r = mH/M, where M = mH +mHe is the total nuclear +mass. +In the present work, we study the two cases of µ = 0.8mn and µ = 1.5mn. We +vary the mass ratio from 0 to 1, meaning that the nuclear center of mass moves from +the helium nucleus to the hydrogen nucleus, see the illustration in figure 1. For fixed +reduced mass, both extreme values of r correspond to infinite total mass. Using a one- +dimensional model of HeH+, we investigate the laser-induced dissociation as a function +of the mass ratio. Our central result is that a strong suppression of dissociation is found +for values of the mass ratio where the electric dipoles are small. We analyze this effect +using both quantum-mechanical and classical simulations. +We note in passing that two isotopologues of the carbon monoxide ion, namely +12C18O+ and 13C16O+, have almost equal reduced masses and they might serve as +another set of example systems for future investigations. +2. Methods +2.1. Electron-nuclear non-Born-Oppenheimer time-dependent Schrödinger equation +We apply a one-dimensional single-active-electron non-Born-Oppenheimer model and +solve the time-dependent Schrödinger equation (TDSE) [17, 18]. +This model covers +two degrees of freedom: electronic motion along the molecular axis described by the +electron coordinate x (electron position relative to the nuclear center of mass) and +the nuclear motion described by the internuclear distance R. A softcore potential for +the electron-nuclear interaction is chosen such that for frozen nuclei, the two lowest +potential-energy curves match the literature values [19, 20]. Previously, this model has +been applied to various problems, including comparisons with experimental data and the +control of dissociation and ionization with two-color fields [17, 18]. The wave function +is represented on a grid with 2048 grid points spaced by 0.05 a.u. along the R-axis and +4096 grid points spaced by 0.2 a.u. along the x-axis. The time step for the propagation is +0.02 a.u. The wave function is propagated using the split-operator method [21] and the + +Mass-ratio dependent strong-field dissociation of helium hydride isotopologues +4 +initial states before interaction with the external field are calculated as eigenstates of the +real-time evolution operator [22]. The real-time evolution starts from the ground state. +The laser pulse is modelled by defining a vector potential A(t) with a cos2 envelope, +A(t) = E0 +ω cos2(πt/T) sin(ωt), +(1) +where T = TFWHM/0.3641 and TFWHM is the full width at half maximum of the intensity, +chosen as TFWHM ≈ 50 fs. The pulse is linearly polarized along the molecular axis. The +vector potential determines the electric field E(t) as E(t) = − ˙A(t). The dissociation +yield is calculated from the wave function at the end of the time evolution by first +projecting out all bound states followed by projection onto electronic eigenstates so +that dissociation into different electronic channels can be distinguished. +2.2. Born-Oppenheimer TDSE +While the non-Born-Oppenheimer model allows us to describe arbitrary electronic +excitations and even ionization, often this is not needed. +For comparison and as a +simpler model, we apply the Born-Oppenheimer approximation and solve the TDSE only +for the nuclear wave functions on two coupled potential curves. Although the coupling +between two electronic states is included in the quantum-mechanical simulations, for +which we present results below, we begin by writing the TDSE for the situation when +this coupling is neglected. In this case, the TDSE for the nuclear wave function ψk(R) +on the k-th potential-energy curve Vk reads +i ∂ +∂tψk(R; t) = Hk(t)ψk(R; t), +(2) +Hk(t) = P 2 +2µ + Vk(R) − dk(R)E(t). +(3) +-1 +0 +1 +2 +3 +4 +5 +6 +7 +8 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 +He +H+ +CM +CM +CM +R +d(R) (a.u.) +internuclear distance R (a.u.) +mass ratio r = 0 +mass ratio r = 1 +Figure 1. Left: Dipole coupling d(R) in the electronic ground state for several values +of the mass ratio r between 0 and 1 in steps of 0.1. Right: Changing the mass ratio +moves the nuclear center of mass (CM) from the helium nucleus to the proton. The +masses of the nuclei are indicated by the size of the circles. Note that the total mass +diverges for r → 0 or r → 1. + +---- +-------------. +---------------. +--- +.-------------. +-- +--- +---Mass-ratio dependent strong-field dissociation of helium hydride isotopologues +5 +Here and in the following, atomic units are used if not stated otherwise. The dipole +moments dk(R) are calculated from the model outlined in section 2.1. To this end, +the k-th electronic eigenstate φk(x; R) is calculated for frozen nuclei at the internuclear +distance R. Since the electron coordinate x is defined relative to the nuclear center of +mass, the functions φk(x; R) depend on the mass ratio r via a coordinate shift, +φk(x; R) = φk,r=0 (x + rR; R) +(4) +Therefore, the purely electronic dipole transition moments, defined as +djk(R) = −⟨φj | x | φk⟩(x), +(5) +satisfy +djk(R) = djk(R) +��� +r=0 + rR δjk. +(6) +For the total dipole moments needed in the Born-Oppenheimer Hamiltonian (3), both +the electron dipole and the charged cores must be taken into account [17, 18], +dk(R) = −⟨φk | (κx + λR) | φk⟩(x), +(7) +where κ = (M + 2)/(M + 1) and λ = (mH − mHe)/M = 2r − 1. Hence the dependence +on r can be written explicitly as +dk(R) = κdkk(R) +��� +r=0 + [1 − (2 − κ)r] R. +(8) +The (permanent) ground-state dipole moment d1(R) is simply called d(R) in the +following. This function is shown in figure 1 for various choices of the mass ratio r. +As motivated above, the dipole moment depends strongly on the mass ratio r. +In the two-level Born-Oppenheimer calculations, the light-induced coupling of the +lowest electronic states is included in the TDSE, which reads +i ∂ +∂t +� +ψ1(R; t) +ψ2(R; t) +� += +� +H1(t) +−κd12E(t) +−κd12E(t) +H2(t) +� � +ψ1(R; t) +ψ2(R; t) +� +(9) +with H1, H2 given by (3) and d12(R) defined in (6). Equation (9) is solved by applying +the split-operator scheme on R grids with 2048 grid points spaced by 0.05 a.u., combined +with the matrix exponential for the offdiagonal part of the Hamiltonian matrix. The +time evolution starts from the vibrational ground state of the lowest electronic state. +At the end of the time evolution, all bound states are projected out from ψ1 and the +norm squared of the remaining wave function is the probability for dissociation into the +electronic ground-state. The squared norm of ψ2 is the probability for dissociation into +the first excited state. +2.3. Classical Calculations +Classical trajectory Monte Carlo (CTMC) simulations are done to investigate the +classical anologue of previously described quantum system. +Similar to the Born- +Oppenheimer TDSE simulations, the system is defined as a particle on a Born- +Oppenheimer potential and the classical Hamiltonian reads the same as in (3). Here, + +Mass-ratio dependent strong-field dissociation of helium hydride isotopologues +6 +10-20 +10-16 +10-12 +10-8 +10-4 +100 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 +6HeD+ +3HeT+ +4HeH+ +probability +mass ratio mH / (mH + mHe) +μ = 0 +� � mn, 3436 nm +same with non-BO TDSE +μ = 1 +� � mn, 4575 nm +μ = � +� � mn, 2376 nm +Figure 2. Dissociation probabilities in the electronic ground state calculated from +the electron-nuclear TDSE (blue triangles) and from the two-level nuclear TDSE (all +other curves, 101 data points each). The laser pulse is 50 fs long with 7 × 1013 W/cm2 +peak intensity. The wavelength is chosen to closely match the v = 0 → 1 transition +(violet line, blue triangles and small green circles) or v = 0 → 2 transition (brown +dashed curve). The initial state is v = 0. Data points that correspond to existing +isotopologues of HeH+ are marked. +we consider only a one-level system, i. e. the system is assumed to stay in the electronic +ground state. The time evolution involves solving Newton’s equations of motion, +dP +dt = F(R, t) = − ∂ +∂R +� +V (R) − d(R)E(t) +� +, +(10) +P = µ dR +dt , +(11) +where F(R, t) is the time-dependent force acting on the particle. Above differential +equations represent an initial value problem for which one has to specify initial conditions +for R and P. The initial conditions are sampled from the Wigner distribution of the +vibrational ground-state wave function ψ0(R), +W(R, P) = 1 +2π +� +ψ∗ +0 +� +R + R′ +2 +� +ψ0 +� +R − R′ +2 +� +eiP R′ dR′. +(12) +The propagation of the trajectories is performed using the fourth-order Runge- +Kutta method [23, 24] using adaptive step size with on-the-fly linear interpolation of the +dipole and potential-energy curves along the R-grid. Trajectories reaching R > 100 a.u. +within the duration of the laser pulse are considered as dissociated. For the remaining +trajectories, dissociation is defined as having final total energy above the asymptotic +value of the ground-state potential-energy curve. The dissociation yield is measured by +the number of dissociated trajectories divided by the total number of initial trajectories. +3. Results and discussion +The dissociation yield as a function of the mass ratio r is shown in figure 2. +We +choose laser frequencies to closely match the resonance v = 0 → 1 or v = 0 → 2. + +Mass-ratio dependent strong-field dissociation of helium hydride isotopologues +7 +10-5 +10-3 +10-1 +(a) +0 → 0 +0 → 1 +0 → 2 +0 → 3 +0 → 4 +0 → 5 +10-4 +10-3 +10-2 +10-1 +| +〈 +v +� + +� +(R) +� + +2 +〉 + +(b ) + → 1 +1 → 2 +2 → 3 +3 → 4 +10-3 +10-2 +10-1 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + + +R + +( + + + + + +( +c +mass ratio r = mH / (mH + mHe) + + +5 +  + +! +"#$% +&'* + +, +- +./34 +678 9 +Figure 3. (a) and (b): Vibrational transition matrix elements |⟨v1|d(R)|v2⟩| for some +vibrational transitions v1 → v2 in the electronic ground state with µ = 0.8mn. (a) +Transitions from the vibrational ground state to other vibrational states. (b) First +four transitions in a vibrational ladder-climbing scheme starting in the ground state +v = 0. (c) Derivative of the dipole coupling d(R) at three fixed internuclear distances. +This quantity is proportional to the classical driving force, see text. +(Here v is the vibrational quantum number.) +Despite always matching a resonant +transition, the dissociation yield changes by many orders of magnitude as a function of +r. The agreement between the Born-Oppenheimer nuclear and non-Born-Oppenheimer +electron-nuclear TDSE is very good, indicating that effects from higher electronic states +(beyond the first excited state) are negligible. +There is a notable minimum around +r = 0.8 for the v = 0 → 1 resonance whereas the yield decreases monotonically with r +for the v = 0 → 2 resonance case. +With increasing r, the dipole coupling d(R) monotonically becomes smaller for +most R as can be seen in figure 1. In a very simple picture where HeH+ consists of a +neutral helium atom and a proton, the dipole coupling is d(R) = (1 − r)R. In this case, +increasing r effectively has the same effect as decreasing the amplitude of the electric +field E(t) in the nuclear Hamiltonian H1, see equation (3). The exact value of d(R) +differs somewhat because the ground-state electron is not exactly located at the helium +nucleus, giving rise to the “bump” in d(R). +As a result, the coupling strengths (dipole matrix elements) of some vibrational +transitions show distinct minima as a function of r, see figure 3. +For the series of +vibrational transitions that are necessary for vibrational ladder climbing, i. e. v = 0 → +1 → 2, etc., several minima close to r = 0.8 play together (see figure 3(b)) to create the +structure in the dissociation yield in figure 2. Note that due to the anharmonicity of the +potential, successive transitions between higher vibrational states are not in resonance + +Mass-ratio dependent strong-field dissociation of helium hydride isotopologues +8 +10-16 +10-12 +10-8 +10-4 +100 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 +probability +mass ratio mH / (mH + mHe) +:;ass< +=iss. +B +O +>s +?iss. +non-B +@ +A s +B +C +Ds. +B +E +FGH e +xI +J +non-B +K +Lst e +M +NP +Figure 4. +Dissociation probabilities for molecules with µ = 0.8mn in a 50 fs laser +pulse with 7 × 1013 W/cm2 peak intensity at 3436 nm wavelength. The green circles +show results from classical calculations on the electronic-ground-state potential curve. +The violet solid curve and the blue triangles show the ground-state dissociation yield +from TDSE calculations (same as in figure 2). Additionally, the population of the +first excited electronic state is shown (yellow dashed curve and orange crosses). The +curves without symbols (101 data points each) are calculated by solving the two-level +nuclear TDSE; the blue triangles and orange crosses are results from the electron- +nuclear TDSE. +for the chosen wavelength, thus some states can be skipped and several excitation +pathways to dissociation may be utilized, some even with similar probabilities. +However, excitation to the first excited vibrational state is a gateway for all relevant +dissociation pathways in the v = 0 → 1 resonance case. This gateway is blocked for +a certain mass ratio, leading to substantial suppression of the dissociation yield. Note +that the dissociation yield in the v = 0 → 2 resonance case in figure 2 follows a similar +trend as the v = 0 → 2 curve in figure 3(a). +The dissociation probabilities from classical calculations are shown as green +circles/line together with the corresponding quantum-mechanical results in figure 4. +While the suppression of dissociation is not as strong, the qualitative behaviour is similar +to the TDSE simulation. The classical analogue to the quantum-mechanical explanation +via dipole transition matrix elements is shown in figure 3(c). The force that a classical +particle on the ground-state potential-energy curve experiences from the laser field is +proportional to d′(R). As a function of the mass ratio r, it is roughly linear (see (8)) +and may or may not cross zero, depending on the internuclear distance. Three examples +for this behaviour are shown in figure 3(c). At the equilibrium distance (R = 1.45 a.u.) +and on the particle’s way out to the dissociation continuum it can be trapped when the +coupling force vanishes, giving rise to the drop in dissociation yield around r = 0.75. +By looking at (9) and the definition of d12, it could be expected that the excitation +to the first excited electronic state is independent of r. Instead, we notice that the yield +in the first excited state (lower curve and points in figure 4) qualitatively follows the +ground-state dissociation yield (upper curve and points) and varies over many orders of + +Mass-ratio dependent strong-field dissociation of helium hydride isotopologues +9 +magnitude. The plateau at approximately 10−15 is probably caused by limited numerical +accuracy and is not physically relevant. These numerical results indicate that the main +pathway to the electronically excited state is not the direct electronic excitation from the +initial vibrational state but instead enhanced excitation at larger internuclear distance. +Clearly, the expansion of the molecule to larger internuclear distance happens primarily +when also the dissociation yield is high. Enhanced excitation at certain internuclear +distances is already known for asymmetric molecules [25]. The need for nuclear motion +as the initial step preceding electronic excitation has also been identified in the ionization +channel of HeH+ [17]. +4. Conclusions +In this work, we have presented numerical results on the dissociation of artificial diatomic +molecular ions driven by strong laser pulses. +We have found that the dissociation +probability is highly dependent on the nuclear mass ratio, even when the reduced nuclear +mass is kept constant. For sufficiently long wavelengths, the dissociation yield exhibits +a distinct minimum at a certain mass ratio where the the transition dipole moments +between vibrational states are strongly suppressed. Classically, a similar suppression +occurs because the laser-induced force on the nuclei is small at certain combinations of +mass ratio and internuclear distance. The effect described here is not due a variation +of the reduced mass, which is another important parameter determining the nuclear +motion. The suppression of dissociation can be traced back to the dependence of the +field-molecule coupling on the location of the nuclear center of mass. This dependence +arises because a molecular ion is a charged system, for which the electric dipole depends +on the choice of reference point. +5. 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Math. Phys. 46 435–53 +[25] Kamta +G +L +and +Bandrauk +A +D +2007 +Phys. +Rev. +A +76(5) +053409 +URL +https://link.aps.org/doi/10.1103/PhysRevA.76.053409 + diff --git a/StE3T4oBgHgl3EQfZwqB/content/tmp_files/load_file.txt b/StE3T4oBgHgl3EQfZwqB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b8a63dad083d8acb811d31b135e09365e3ece82e --- /dev/null +++ b/StE3T4oBgHgl3EQfZwqB/content/tmp_files/load_file.txt @@ -0,0 +1,349 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf,len=348 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='04500v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='atom-ph] 11 Jan 2023 Mass-ratio dependent strong-field dissociation of artificial helium hydride isotopologues F Oppermann1, S Mhatre2, S Gräfe2,3, M Lein1 1 Leibniz University Hannover, Institute of Theoretical Physics, Appelstraße 2, 30167 Hannover, Germany 2 Institute of Physical Chemistry, Friedrich Schiller University Jena, Helmholtzweg 4, 07743 Jena, Germany 3 Fraunhofer Institute of Applied Optics and Precision Engineering, Albert-Einstein-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' 7, 07745 Jena, Germany E-mail: lein@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='uni-hannover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='de Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' We study the effect of the nuclear-mass ratio in a diatomic molecular ion on the dissociation dynamics in strong infrared laser pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' A molecular ion is a charged system, in which the dipole moment depends on the reference point and therefore on the position of the nuclear center of mass, so that the laser-induced dynamics is expected to depend on the mass asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Whereas usually both the reduced mass and the mass ratio are varied when different isotopologues are compared, we fix the reduced mass and artificially vary the mass ratio in a model system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' This allows us to separate effects related to changes in the resonance frequency, which is determined by the reduced mass, from those that arise due to the mass asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Numerical solutions of the time-dependent Schrödinger equation are compared with classical trajectory simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' We find that at a certain mass ratio, vibrational excitation is strongly suppressed, which decreases the dissociation probability by many orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Keywords: strong laser fields, helium hydride molecular ion, laser-induced dissociation, time-dependent Schrödinger equation Mass-ratio dependent strong-field dissociation of helium hydride isotopologues 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Introduction Molecules under the influence of an external field can undergo various types of excitation and, in the case of a strong laser field, can be ionized or dissociated [1, 2, 3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' For long wavelengths of the laser field, a large number of photons is needed to overcome the electronic excitation energies and hence, electronic transitions become unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The direct excitation of atomic motion within the electronic ground state, on the other hand, requires (within the electric-dipole approximation) the presence of a non-zero permanent electric dipole or at least a change of the dipole with the geometry of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' For this reason, direct vibrational excitations are dipole-forbidden in homonuclear diatomic molecules due to their strictly vanishing electric dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Heteronuclear diatomic molecules have a dipole that couples directly to the applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The permanent dipole moment and the nuclear masses are the relevant parameters that determine the quantitative amount of vibrational excitation and dissociation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' For neutral diatomic molecules, we are used to the idea that the permanent dipole at a specified internuclear vector has a well-defined value independent of the nuclear masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' For a molecular ion, however, we must take into account the fundamental statement that the dipole moment of a charged system depends on the choice of reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' According to the separation into center-of-mass motion and relative motion, the relevant dipoles for vibrational excitation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=', excitation of the relative motion) must be calculated with the center of nuclear mass as the reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Thus, the dipole moments of molecular ions depend on the nuclear masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' In a diatomic system, the reduced nuclear mass is, besides the dipole, the other crucial parameter that determines the nuclear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The reduced mass determines the vibrational energy levels and therefore the values of the resonant transition frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Here it is important to note that the dipole moment depends on the nuclear masses even when the reduced mass is kept constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Isotope effects in photodissociation processes have always been a matter of interest, and many different molecular species have been studied, ranging from simple diatomic molecules [6, 7] to polyatomic organic molecules [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Nevertheless, the effect of the mass-dependent electric dipoles has not been isolated since one requires a charged system, and furthermore, the separation from changes in the reduced mass is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' It would be interesting to observe the isotopologue dependence of laser-induced dissociation of a molecular ion at fixed reduced mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Unfortunately, there are few realistic target systems for such a purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' In the present work, we consider artificial isotopologues of the HeH+ molecular ion, which can be considered the simplest heteronuclear molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' HeH+ can be prepared in the laboratory [10] and it has recently been observed in interstellar space [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' It has already served as an asymmetric polar benchmark system in a number of studies [12, 13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Its isotopologues, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' 4HeH+, 4HeD+, 3HeH+, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' possess the same electronic configuration but they differ in both total and reduced nuclear mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Most of the isotopologues that could in principle be constructed from the real isotopes of He and H differ in reduced mass, but there are two examples, 3HeT+ and 6HeD+, with (approximately) the same reduced mass—albeit not Mass-ratio dependent strong-field dissociation of helium hydride isotopologues 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Properties of selected isotopologues of HeH+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' isotopologue total mass reduced mass mass ratio r = mH/M 4HeH+ 5mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='8mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='2 4HeD+ 6mn 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='33mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='33 3HeT+ 6mn 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='5mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='5 6HeD+ 8mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='25 easily experimentally available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' (6He has a half-life time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='8 s [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=') In Table 1, we show the properties of selected isotopologues of HeH+, where, for simplicity, protons and neutrons are idealized as having equal masses mn = 1837 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' and the binding energy (mass defect) of the nuclei is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The reduced mass is given by µ = mHmHe/M and we define the mass ratio r as r = mH/M, where M = mH +mHe is the total nuclear mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' In the present work, we study the two cases of µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='8mn and µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='5mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' We vary the mass ratio from 0 to 1, meaning that the nuclear center of mass moves from the helium nucleus to the hydrogen nucleus, see the illustration in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' For fixed reduced mass, both extreme values of r correspond to infinite total mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Using a one- dimensional model of HeH+, we investigate the laser-induced dissociation as a function of the mass ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Our central result is that a strong suppression of dissociation is found for values of the mass ratio where the electric dipoles are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' We analyze this effect using both quantum-mechanical and classical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' We note in passing that two isotopologues of the carbon monoxide ion, namely 12C18O+ and 13C16O+, have almost equal reduced masses and they might serve as another set of example systems for future investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Electron-nuclear non-Born-Oppenheimer time-dependent Schrödinger equation We apply a one-dimensional single-active-electron non-Born-Oppenheimer model and solve the time-dependent Schrödinger equation (TDSE) [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' This model covers two degrees of freedom: electronic motion along the molecular axis described by the electron coordinate x (electron position relative to the nuclear center of mass) and the nuclear motion described by the internuclear distance R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' A softcore potential for the electron-nuclear interaction is chosen such that for frozen nuclei, the two lowest potential-energy curves match the literature values [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Previously, this model has been applied to various problems, including comparisons with experimental data and the control of dissociation and ionization with two-color fields [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The wave function is represented on a grid with 2048 grid points spaced by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='05 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' along the R-axis and 4096 grid points spaced by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' along the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The time step for the propagation is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='02 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The wave function is propagated using the split-operator method [21] and the Mass-ratio dependent strong-field dissociation of helium hydride isotopologues 4 initial states before interaction with the external field are calculated as eigenstates of the real-time evolution operator [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The real-time evolution starts from the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The laser pulse is modelled by defining a vector potential A(t) with a cos2 envelope, A(t) = E0 ω cos2(πt/T) sin(ωt), (1) where T = TFWHM/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='3641 and TFWHM is the full width at half maximum of the intensity, chosen as TFWHM ≈ 50 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The pulse is linearly polarized along the molecular axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The vector potential determines the electric field E(t) as E(t) = − ˙A(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The dissociation yield is calculated from the wave function at the end of the time evolution by first projecting out all bound states followed by projection onto electronic eigenstates so that dissociation into different electronic channels can be distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Born-Oppenheimer TDSE While the non-Born-Oppenheimer model allows us to describe arbitrary electronic excitations and even ionization, often this is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' For comparison and as a simpler model, we apply the Born-Oppenheimer approximation and solve the TDSE only for the nuclear wave functions on two coupled potential curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Although the coupling between two electronic states is included in the quantum-mechanical simulations, for which we present results below, we begin by writing the TDSE for the situation when this coupling is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' In this case, the TDSE for the nuclear wave function ψk(R) on the k-th potential-energy curve Vk reads i ∂ ∂tψk(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' t) = Hk(t)ψk(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' t), (2) Hk(t) = P 2 2µ + Vk(R) − dk(R)E(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' (3) 1 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 He H+ CM CM CM R d(R) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=') internuclear distance R (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=') mass ratio r = 0 mass ratio r = 1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Left: Dipole coupling d(R) in the electronic ground state for several values of the mass ratio r between 0 and 1 in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Right: Changing the mass ratio moves the nuclear center of mass (CM) from the helium nucleus to the proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The masses of the nuclei are indicated by the size of the circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Note that the total mass diverges for r → 0 or r → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' ---- -------------.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' ---------------.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' --- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='-------------.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' -- --- ---Mass-ratio dependent strong-field dissociation of helium hydride isotopologues 5 Here and in the following, atomic units are used if not stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The dipole moments dk(R) are calculated from the model outlined in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' To this end, the k-th electronic eigenstate φk(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' R) is calculated for frozen nuclei at the internuclear distance R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Since the electron coordinate x is defined relative to the nuclear center of mass, the functions φk(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' R) depend on the mass ratio r via a coordinate shift, φk(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' R) = φk,r=0 (x + rR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' R) (4) Therefore, the purely electronic dipole transition moments, defined as djk(R) = −⟨φj | x | φk⟩(x), (5) satisfy djk(R) = djk(R) ��� r=0 + rR δjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' (6) For the total dipole moments needed in the Born-Oppenheimer Hamiltonian (3), both the electron dipole and the charged cores must be taken into account [17, 18], dk(R) = −⟨φk | (κx + λR) | φk⟩(x), (7) where κ = (M + 2)/(M + 1) and λ = (mH − mHe)/M = 2r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Hence the dependence on r can be written explicitly as dk(R) = κdkk(R) ��� r=0 + [1 − (2 − κ)r] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' (8) The (permanent) ground-state dipole moment d1(R) is simply called d(R) in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' This function is shown in figure 1 for various choices of the mass ratio r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' As motivated above, the dipole moment depends strongly on the mass ratio r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' In the two-level Born-Oppenheimer calculations, the light-induced coupling of the lowest electronic states is included in the TDSE, which reads i ∂ ∂t � ψ1(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' t) ψ2(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' t) � = � H1(t) −κd12E(t) −κd12E(t) H2(t) � � ψ1(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' t) ψ2(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' t) � (9) with H1, H2 given by (3) and d12(R) defined in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Equation (9) is solved by applying the split-operator scheme on R grids with 2048 grid points spaced by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='05 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=', combined with the matrix exponential for the offdiagonal part of the Hamiltonian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The time evolution starts from the vibrational ground state of the lowest electronic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' At the end of the time evolution, all bound states are projected out from ψ1 and the norm squared of the remaining wave function is the probability for dissociation into the electronic ground-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The squared norm of ψ2 is the probability for dissociation into the first excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Classical Calculations Classical trajectory Monte Carlo (CTMC) simulations are done to investigate the classical anologue of previously described quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Similar to the Born- Oppenheimer TDSE simulations, the system is defined as a particle on a Born- Oppenheimer potential and the classical Hamiltonian reads the same as in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Here, Mass-ratio dependent strong-field dissociation of helium hydride isotopologues 6 10-20 10-16 10-12 10-8 10-4 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='8 1 6HeD+ 3HeT+ 4HeH+ probability mass ratio mH / (mH + mHe) μ = 0 � � mn, 3436 nm same with non-BO TDSE μ = 1 � � mn, 4575 nm μ = � � � mn, 2376 nm Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Dissociation probabilities in the electronic ground state calculated from the electron-nuclear TDSE (blue triangles) and from the two-level nuclear TDSE (all other curves, 101 data points each).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The laser pulse is 50 fs long with 7 × 1013 W/cm2 peak intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The wavelength is chosen to closely match the v = 0 → 1 transition (violet line, blue triangles and small green circles) or v = 0 → 2 transition (brown dashed curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The initial state is v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Data points that correspond to existing isotopologues of HeH+ are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' we consider only a one-level system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' the system is assumed to stay in the electronic ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The time evolution involves solving Newton’s equations of motion, dP dt = F(R, t) = − ∂ ∂R � V (R) − d(R)E(t) � , (10) P = µ dR dt , (11) where F(R, t) is the time-dependent force acting on the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Above differential equations represent an initial value problem for which one has to specify initial conditions for R and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The initial conditions are sampled from the Wigner distribution of the vibrational ground-state wave function ψ0(R), W(R, P) = 1 2π � ψ∗ 0 � R + R′ 2 � ψ0 � R − R′ 2 � eiP R′ dR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' (12) The propagation of the trajectories is performed using the fourth-order Runge- Kutta method [23, 24] using adaptive step size with on-the-fly linear interpolation of the dipole and potential-energy curves along the R-grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Trajectories reaching R > 100 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' within the duration of the laser pulse are considered as dissociated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' For the remaining trajectories, dissociation is defined as having final total energy above the asymptotic value of the ground-state potential-energy curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The dissociation yield is measured by the number of dissociated trajectories divided by the total number of initial trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Results and discussion The dissociation yield as a function of the mass ratio r is shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' We choose laser frequencies to closely match the resonance v = 0 → 1 or v = 0 → 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Mass-ratio dependent strong-field dissociation of helium hydride isotopologues 7 10-5 10-3 10-1 (a) 0 → 0 0 → 1 0 → 2 0 → 3 0 → 4 0 → 5 10-4 10-3 10-2 10-1 | 〈 v � � (R) � 2 〉 (b ) → 1 1 → 2 2 → 3 3 → 4 10-3 10-2 10-1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='8 1 \x0e R \x0f ( \x10 \x11 \x12 \x13 \x14 ( c\x15 mass ratio r = mH / (mH + mHe) \x16 \x17 \x18\x19\x1a5 \x1b !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' "#$% &\'* + , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='/34 678 9 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' (a) and (b): Vibrational transition matrix elements |⟨v1|d(R)|v2⟩| for some vibrational transitions v1 → v2 in the electronic ground state with µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='8mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' (a) Transitions from the vibrational ground state to other vibrational states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' (b) First four transitions in a vibrational ladder-climbing scheme starting in the ground state v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' (c) Derivative of the dipole coupling d(R) at three fixed internuclear distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' This quantity is proportional to the classical driving force, see text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' (Here v is the vibrational quantum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=') Despite always matching a resonant transition, the dissociation yield changes by many orders of magnitude as a function of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The agreement between the Born-Oppenheimer nuclear and non-Born-Oppenheimer electron-nuclear TDSE is very good, indicating that effects from higher electronic states (beyond the first excited state) are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' There is a notable minimum around r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='8 for the v = 0 → 1 resonance whereas the yield decreases monotonically with r for the v = 0 → 2 resonance case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' With increasing r, the dipole coupling d(R) monotonically becomes smaller for most R as can be seen in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' In a very simple picture where HeH+ consists of a neutral helium atom and a proton, the dipole coupling is d(R) = (1 − r)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' In this case, increasing r effectively has the same effect as decreasing the amplitude of the electric field E(t) in the nuclear Hamiltonian H1, see equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The exact value of d(R) differs somewhat because the ground-state electron is not exactly located at the helium nucleus, giving rise to the “bump” in d(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' As a result, the coupling strengths (dipole matrix elements) of some vibrational transitions show distinct minima as a function of r, see figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' For the series of vibrational transitions that are necessary for vibrational ladder climbing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' v = 0 → 1 → 2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=', several minima close to r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='8 play together (see figure 3(b)) to create the structure in the dissociation yield in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Note that due to the anharmonicity of the potential, successive transitions between higher vibrational states are not in resonance Mass-ratio dependent strong-field dissociation of helium hydride isotopologues 8 10-16 10-12 10-8 10-4 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='8 1 probability mass ratio mH / (mH + mHe) :;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='ass< =iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' B O >s ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' non-B @ A s B C Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' B E FGH e xI J non-B K Lst e M NP Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Dissociation probabilities for molecules with µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='8mn in a 50 fs laser pulse with 7 × 1013 W/cm2 peak intensity at 3436 nm wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The green circles show results from classical calculations on the electronic-ground-state potential curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The violet solid curve and the blue triangles show the ground-state dissociation yield from TDSE calculations (same as in figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Additionally, the population of the first excited electronic state is shown (yellow dashed curve and orange crosses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The curves without symbols (101 data points each) are calculated by solving the two-level nuclear TDSE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' the blue triangles and orange crosses are results from the electron- nuclear TDSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' for the chosen wavelength, thus some states can be skipped and several excitation pathways to dissociation may be utilized, some even with similar probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' However, excitation to the first excited vibrational state is a gateway for all relevant dissociation pathways in the v = 0 → 1 resonance case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' This gateway is blocked for a certain mass ratio, leading to substantial suppression of the dissociation yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Note that the dissociation yield in the v = 0 → 2 resonance case in figure 2 follows a similar trend as the v = 0 → 2 curve in figure 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The dissociation probabilities from classical calculations are shown as green circles/line together with the corresponding quantum-mechanical results in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' While the suppression of dissociation is not as strong, the qualitative behaviour is similar to the TDSE simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The classical analogue to the quantum-mechanical explanation via dipole transition matrix elements is shown in figure 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The force that a classical particle on the ground-state potential-energy curve experiences from the laser field is proportional to d′(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' As a function of the mass ratio r, it is roughly linear (see (8)) and may or may not cross zero, depending on the internuclear distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Three examples for this behaviour are shown in figure 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' At the equilibrium distance (R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='45 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=') and on the particle’s way out to the dissociation continuum it can be trapped when the coupling force vanishes, giving rise to the drop in dissociation yield around r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' By looking at (9) and the definition of d12, it could be expected that the excitation to the first excited electronic state is independent of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Instead, we notice that the yield in the first excited state (lower curve and points in figure 4) qualitatively follows the ground-state dissociation yield (upper curve and points) and varies over many orders of Mass-ratio dependent strong-field dissociation of helium hydride isotopologues 9 magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The plateau at approximately 10−15 is probably caused by limited numerical accuracy and is not physically relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' These numerical results indicate that the main pathway to the electronically excited state is not the direct electronic excitation from the initial vibrational state but instead enhanced excitation at larger internuclear distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Clearly, the expansion of the molecule to larger internuclear distance happens primarily when also the dissociation yield is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Enhanced excitation at certain internuclear distances is already known for asymmetric molecules [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The need for nuclear motion as the initial step preceding electronic excitation has also been identified in the ionization channel of HeH+ [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Conclusions In this work, we have presented numerical results on the dissociation of artificial diatomic molecular ions driven by strong laser pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' We have found that the dissociation probability is highly dependent on the nuclear mass ratio, even when the reduced nuclear mass is kept constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' For sufficiently long wavelengths, the dissociation yield exhibits a distinct minimum at a certain mass ratio where the the transition dipole moments between vibrational states are strongly suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' Classically, a similar suppression occurs because the laser-induced force on the nuclei is small at certain combinations of mass ratio and internuclear distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The effect described here is not due a variation of the reduced mass, which is another important parameter determining the nuclear motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' The suppression of dissociation can be traced back to the dependence of the field-molecule coupling on the location of the nuclear center of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' This dependence arises because a molecular ion is a charged system, for which the electric dipole depends on the choice of reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfZwqB/content/2301.04500v1.pdf'} +page_content=' 5.' metadata={'source': 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b/VdE3T4oBgHgl3EQfbApG/content/tmp_files/2301.04511v1.pdf.txt @@ -0,0 +1,1171 @@ +1 +Federated Learning and Blockchain-enabled +Fog-IoT Platform for Wearables in Predictive +Healthcare +Marc Jayson Baucas, Student Member, IEEE, Petros Spachos, Senior Member, IEEE, and +Konstantinos N. Plataniotis, Fellow, IEEE +Abstract—Over the years, the popularity and usage of wear- +able Internet of Things (IoT) devices in several healthcare +services are increased. Among the services that benefit from the +usage of such devices is predictive analysis, which can improve +early diagnosis in e-health. However, due to the limitations +of wearable IoT devices, challenges in data privacy, service +integrity, and network structure adaptability arose. To address +these concerns, we propose a platform using federated learning +and private blockchain technology within a fog-IoT network. +These technologies have privacy-preserving features securing +data within the network. We utilized the fog-IoT network’s +distributive structure to create an adaptive network for wearable +IoT devices. We designed a testbed to examine the proposed +platform’s ability to preserve the integrity of a classifier. Accord- +ing to experimental results, the introduced implementation can +effectively preserve a patient’s privacy and a predictive service’s +integrity. We further investigated the contributions of other +technologies to the security and adaptability of the IoT network. +Overall, we proved the feasibility of our platform in addressing +significant security and privacy challenges of wearable IoT +devices in predictive healthcare through analysis, simulation, and +experimentation. +Index Terms—Machine learning, Data privacy, Predictive +models, Distributed systems, Health care services, Platforms, +Testbed, Health informatics, Fog network, Security, Private +Blockchain, Privacy, Scalability, Internet of Things. +I. INTRODUCTION +T +HE usage of wearable Internet of Things (IoT) devices +in healthcare is rising. Due to their availability and sens- +ing capability, these devices collect physiological data from +patients and provide real-time diagnosis [1]. Wearable IoT de- +vices have caused remote healthcare to make a paradigm shift +into predictive diagnosis and reliable early detection. The data +collected by these devices partnered with different learning +techniques aid in predictive healthcare services. Doctors can +analyze data such as their patient’s activities and accurately +predict anomalies and threats against their health [2]. They can +also prescribe treatments for preventing and addressing these +detected concerns. However, this breakthrough has limitations +This work was supported in part by the Natural Sciences and Engineering +Research Council (NSERC) of Canada. +M. Baucas and P. Spachos are with the School of Engineering, University +of Guelph, Guelph, ON, N1G2W1, Canada. (e-mail: baucas@uoguelph.ca; +petros@uoguelph.ca). +K. Plataniotis is with the Department of Electrical and Computer En- +gineering, University of Toronto, Toronto, ON, M5S3G4, Canada. (e-mail: +kostas@ece.utoronto.ca) +in its technology. Challenge within a network that employs +wearable IoT devices cause impasses in predictive healthcare. +An open issue for wearable IoT devices in predictive +healthcare is the amount of data it needs to be effective. A +large amount of personal data is collected, resulting in security +and privacy concerns due to the nature of the data used for +analysis [3]. The wearable devices’ limitations on processing +capabilities lead to vulnerabilities and potential leakages in +sensitive patient information [4]. Another issue is the integrity +and reliability of the service [5]. Structuring healthcare to +prioritize certain aspects can cause trade-offs in others. Service +integrity is crucial for this field in remote healthcare that relies +on wearable data accuracy and predictive model precision. +One more issue is the adaptability of the network that deploys +and serves these predictive healthcare services [6]. Wearable +IoT device standardization is a significant concern to IoT +networks due to the heterogeneity it introduces. This diversity +results in demands for continuous maintenance and updates +to the medical server to ensure that it is up to date with +every newly introduced wearable IoT device. As a result, +concerns about adaptability limit the healthcare network from +fully remaining relevant and sustainable over long stretches +of its service. +In this work, we propose a fog-IoT platform to address +these issues. We use federated learning to preserve patient +data privacy and the integrity of the network’s predictive +services [7]. Also, we incorporate blockchain technology to +address the security issues in wearable IoT devices through its +access control and cryptographic structure [8], [9]. Finally, we +combine these technologies within a fog-based IoT architec- +ture to enforce decentralized servers and resource reallocation +to improve the adaptability and sustainability of the overall +network [10]. The main contributions of this work are: +• We present a fog-based IoT platform using federated +learning and blockchain technology to preserve patient +data privacy and improve the security of data within the +network. +• We designed a testbed that simulates and evaluates the +proposed implementation. We used model accuracy to +observe the platform’s ability to preserve the integrity of +a predictive service. +The rest of this paper is organized as follows. A discussion +on the background of our study and a brief literature review +are in Section II. Our proposed design and the methodology +arXiv:2301.04511v1 [cs.LG] 11 Jan 2023 + +2 +followed are in Section III. The presentation of a developed +testbed and a discussion of the results are in Section IV. +Finally, our conclusions are in Section V. +II. BACKGROUND AND RELATED WORKS +We provide a brief discussion on the relevant works in +blockchain technology and federated learning implementa- +tions for wearable IoT devices in healthcare. +A. Benefits of Using Wearable IoT Devices in Healthcare +Wearable IoT devices can form a network of sensing +devices that collect data from points of interest for predic- +tive analysis. In predictive healthcare, these devices collect +physiological information from patients for better clinical +decisions and to provide a prediction. Each obtained statistic +reflects the status of their health, which can identify future +issues and potential risks. A benefit of using wearable IoT +devices in healthcare is to reduce hospital congestion and +medical examination costs via remote diagnoses and long- +distance patient monitoring. In [11], they introduce a wearable +Tele-ECG system for heart rate monitoring. The proposed +design merges the latest technologies in textile electrodes +(TE), Bluetooth low energy (BLE), and smartphones to create +a portable means of monitoring and evaluating the condition +of a patient’s heart for potential anomalies. With features such +as geographical tracking, medical history storage, and remote +patient monitoring, the system establishes a light and cost- +effective alternative for patients suffering from heart issues +that need constant observation and evaluation. +Another advantage of using wearable IoT devices in pre- +dictive healthcare is improving the quality of early disease +and fault detection in medical centres. In [12], they present a +review of various biomedical IoT devices that raise the quality +of diagnosis in healthcare services. The survey includes wear- +able sensors as one of the leading technologies that enable +advancements in predictive healthcare IoT. Services are now +made remote and analytic. Patients can use technologies and +wearable sensors that can help give doctors early information +on potential causes and triggers of health complications. As a +result, there is improvement in predictive anomaly detection +without constraints in geographical location and data resource +availability. Also, these devices widen the scope of medical +centres toward their patients. Doctors can use trend and +behaviour analysis to pinpoint anomalies in a patient’s health. +At the same time, medical professionals can evaluate diseases +with improved precision due to leveraged real-time and early +detection [13]. +B. Challenges of Using Wearable IoT Devices in Healthcare +Although the usage of wearable IoT devices shows potential +to raise the quality of services in healthcare, they also pose +the following concerns. +1) Network Security and Data Privacy: Introducing wear- +able devices to collect patient data adds more endpoints to +the server of the healthcare service. Data collection and mon- +itoring are convenient due to these remote services. However, +introducing more devices introduces more vulnerabilities to +the network. Increasing the endpoints increases the potential +areas where malicious users can attack and steal data. As a +result, there is a greater demand for better security as the +network grows. Also, privacy becomes a concern due to the +sensitive information transmitted from the wearable device +to the server. Therefore, network security and data privacy +concerns grow as the network expands with more wearable +devices. +Different strategies and technologies have been proposed +to address this issue [14], [15]. In [14], they present a multi- +keyword search mechanism to preserve data privacy within +the IoT network. They aim to strengthen the encryption of +the information transmitted from the patient to the medical +centre. This strategy achieves forward privacy preservation, +which results in a more guaranteed security system for users. +Another example that aims for privacy preservation is in [15]. +They use deep learning to secure the network by separating +private from raw data. This strategy results in minimizing the +chances of leaking sensitive information. +Our approach takes key strengths from these two strategies. +We take the ability of cryptographic techniques to reinforce +data transmissions and combine them with the adaptability +of machine learning techniques to create a secure means of +preserving patient data. As a result, we chose blockchain +technology for its strong encryption and federated learning +for its adaptive ability to improve data privacy while keeping +its trained models optimized. +2) Data Integrity and Precision: An advantage of using +wearable IoT devices in healthcare is the real-time diagnosis +and early detection of illness and medical anomalies within +a patient. However, this can be affected by the quality of the +sensor and the data processing scheme behind the service. +Potential misdiagnosis is high if the sensor or the evaluation +method is not up to standard. Another source of error could +come from how the data is collected and stored on the server. +Preserving physiological information integrity obtained from +the wearable devices is needed as it travels from patient to +server. It ensures that the data is recent and precise to the +current status of the source. Also, the data handling during +aggregation should be carried out with great precision so that +the evaluation of the patient is accurate. A service that can not +collect, transmit, and process the data correctly and efficiently +will result in inaccurate results. Therefore, the integrity and +precision of information a concern that needs addressing for +a healthcare service that uses wearable IoT devices to be +effective. +Approaches that improve the integrity of sensed data from +IoT devices in healthcare services are proposed in [16], [17]. +In [16], a novel data aggregator for efficient and secure +analysis in IoT monitoring is presented. Its scheme involves +a cryptographic accumulator that securely collects data from +wearable IoT devices. Another example is in [17]. This work +highlights the advantages of integrating blockchains with IoT. +They aim to enhance the performance of healthcare services +that use wearable IoT devices by using the security advantages +of blockchain technology. Similar to the approach in [16], both +strategies focus on the ability of cryptographic technologies + +3 +to ensure the accuracy of sensed data within the IoT network. +Our proposed approach differs as it incorporates federated +learning for a more secure data aggregator. We combine it +with decentralized organizations provided by blockchain and +create a platform that ensures a secure medium for accurate +data analysis in healthcare. +3) Network Structure Adaptability and Flexibility: Wear- +able IoT devices introduce different sensors and technologies +that collect and process data. Due to their multiple advantages, +healthcare services should include them in their systems. How- +ever, innovations and improvements are iterative as changes +arise often. Although this is a sign of technological growth +and advancements in the healthcare sector, it dictates the +pace at which current infrastructures need to keep up. Device +diversity has always been a concern for IoT networks due to +the need for ongoing standardization. As this new technology +is still in development, services that use it should be able to +follow the development process. It is the same for wearable +IoT devices used in healthcare. The demand for technologies +that detect each one is high as new diseases and virus strains +often appear. However, every new technology introduced a +need for the service to adapt. Networks that are less flexible +result in losing their resources. Some even end up losing their +service and infrastructure altogether. Therefore, there is a need +for adaptability and flexibility in the structure of healthcare +networks for the continuously evolving nature of wearable +IoT devices. +Some approaches focus on improving the flexibility of +wearable devices to tackle this issue. In [18], they present an +innovative wrist-worn prototype for patient monitoring, which +also functions as an IoT gateway. Some sensors in remote +healthcare focus on sensing the environment around the patient +to ensure that the condition of their surrounding is beneficial to +their health. Through it, ambient sensing devices can connect +to the healthcare network. As a result, it sets up a platform that +can easily integrate and synchronize other devices under it. +In [19], they present a wearable patch that functions similarly +to the prototype in [18] by creating a flexible IoT gateway +for connecting wearable devices. Instead of ambient sensors, +their portal focuses on ECGs and PPGs. +Our approach is different as we aim to standardize devices +through the fog by moving analysis closer to the edge. As a +result, we focus on data management standardization instead +of device standardization due to the diversity of wearable +devices. Instead of specialization through prototypes, we plan +to use fog-IoT paired with decentralized technologies such as +blockchain and federated learning to develop a modular IoT +network for wearable devices in healthcare. +C. Federated Learning for Wearable IoT Devices +Federated learning is a machine learning technique that +takes a distributed approach in training its models [20]. It uses +its decentralized strategy to utilize global knowledge collected +from its clients. In IoT, federated learning has two compo- +nents; the client and the server [21]. Its flow of operations for +federated learning starts with each client training their model +using their raw data. Then, the server aggregates and compiles +the resulting models into a global model that it redistributes +to each client’s use. As a result, each client receives the most +optimal model given global knowledge gathered through local +training. This procedure is ongoing, and the global model is +updated every time the client provides new knowledge. +Federated learning is well-known for its capability of effec- +tively preserving the privacy of data [22]. Requiring clients to +transmit raw data directly to the server before it is analyzed +can cause vulnerabilities. Without a strongly reinforced net- +work, malicious attackers aiming to steal or tamper can target +the data. Federated learning protects raw data by creating +a strategy that moves the analysis locally. There are lesser +avenues for privacy leakages since the server only cares about +the resulting trained model from each client [23]. As a result, +this structure has an improved trusted architecture compared +to standard network arrangements. Aside from its security +advantages, federated learning also improves the sustainability +of IoT networks. It provides a dynamic learning strategy +that keeps global knowledge for services up to date [24]. +Commonly, the server aggregates all the data first before it +uses it to train the global model. Federated learning allows +new correlations and behaviour changes from collected data +to be detected by the server earlier. A fully centralized scheme +is slowed down and saturated due to the volume of the data it +uses to train its models. Federated learning can reduce training +time by running it locally and in parallel. Also, there is a +significant reduction in the data size as each client trains its +model. +Healthcare services that involve wearable IoT devices re- +volve around continuous data sensing, learning, and analysis. +Federated learning can provide a sustainable and decentralized +scheme to optimize these services in healthcare [25]. We +chose to use this technology in our platform due to all the +improvements it can contribute to IoT services, specifically +healthcare. Our proposed design is different because we aim +to reinforce federated learning with the security provided by +blockchains. Also, the foundation of our network will be a +fog IoT network. The diversity of wearable IoT devices and +their limited processing capabilities is a constraint. However, +we can overcome this by moving the training to the fog. This +arrangement still allows training locally while considering the +processing limits of wearable IoT devices. The result is a +decentralized IoT network that can maximize the potential +of federated learning in keeping data analysis in healthcare +services secure and sustainable. +D. Blockchain for Wearable IoT Devices +A blockchain is a cryptographic ledger that provides a dis- +tributed service for information storage and security [26]. This +decentralized control establishes a trusted system that will +only grant access to users acknowledged by the blockchain +ledger [27]. This feature protects the data from external or +unwanted modifications and makes it immutable. For wearable +IoT devices in healthcare, the network can use blockchains as +a tamper-proof database for patient data storage. +Its unique structure addresses different security issues in +IoT networks. In [28], they use the unique consent mechanism + +4 +of blockchains to secure user data privacy in wearable fitness +devices. They aim to prove the feasibility of the trustwor- +thiness of blockchains when fortifying the data privacy of +wearable IoT device data. Another proposed use for the access +control of blockchains is in [29]. They developed a lightweight +authentication scheme that classifies mobile devices and dif- +ferentiates them from fake data injections or illegitimate users. +Their designs show the potential of blockchain technology +in securing the healthcare server and the monitoring devices +used by its services. In [30], they use a different feature of +blockchains in reinforcing their healthcare service that uses +wearable IoT devices. They integrate the encryption model of +blockchains to improve the security of the IoT network. Their +design creates a searchable encryption technique that assists in +securing collected COVID-19 data. It shows the effectiveness +of blockchain architecture in helping protect the healthcare +IoT network from several security threats like malicious data +injection and hijacking sessions. +Our design differs from the previously presented imple- +mentations because we aim to use a private blockchain [31]. +Public blockchains use a reward-based protocol called “proof- +of-work” (PoW) in granting access to their devices [32]. +Although this protocol is secure, its processing requirement +is a caveat for wearable IoT devices in healthcare. In [33], +we highlighted how some wearable IoT devices are incapable +of adapting to this authentication structure due to their low- +cost and low-end designs. Therefore, we need a different +blockchain architecture that can cater to the design nature +of wearable devices. Private blockchains function differently. +They incorporate a more trust-based protocol. This approach +turns the blockchain into a hyperledger that can regulate its +devices based on a list of trusted devices defined by the +network owner. Healthcare IoT networks can benefit from this +approach more since the need for high processing power for +their wearable devices is reduced. As a result, the types of +IoT-based wearable devices they can incorporate into their +services do not limit the healthcare network [34]. +III. DESIGN AND METHODOLOGY +The following is a discussion of our proposed platform, +including a description of the design and the different com- +ponents. +A. Architecture Overview +We propose a fog-IoT-based approach to secure data ex- +change from wearable IoT devices used for predictive health- +care services. We aim to improve the overall structure’s service +integrity and flexibility. To further enhance the distributive +organization of our network, we chose to use federated learn- +ing. With its ability to aggregate learning models, we can +sort our data and keep private information secure. Also, it can +reduce the sources of leakage since each client is only required +to send their locally trained model to the server. Integrating +federated learning improves the sustainability of the predictive +healthcare service by providing a system that can organize +the model data and provide a global model that represents +the overall knowledge obtained through all the local training. +Fig. 1: Network arrangement for the proposed fog-based +platform for wearable IoT devices. +The result is a scalable design that introduces flexibility by +keeping the network distributed while ensuring that it does +not impede the learning and analysis of the service. +Next, +we +combine +federated +learning +with +private +blockchain technology for more secure client authorization. +Also, it is treated as a hyperledger to store the IDs of +each client device and locally trained model. Due to its +immutability, tracking and logging each trained model for +version control and debugging improves the data analysis and +learning aspect of healthcare networks and their services. +Commercial wearable devices are diverse. Some are low- +end by design. To address this, we moved the process of +locally training the model from the edge of the network to the +fog. We can reallocate the training process by providing an +intermediary fog device. Usually, training takes a large portion +of processor capacity. Unfortunately, not every wearable IoT +device can do it effectively. For our platform to be able and +handle a diverse pool of wearable devices, we offloaded the +processing required to a fog device. +The proposed network arrangement is shown in Fig. 1. First, +the fog device collects all sensed data from each wearable +IoT device and trains the model locally. Each one sends its +local model to the server for storage and tracking. Next, the +server aggregates all the knowledge from the local models and +generates a global model. Then, the server sends this model +to each wearable IoT device through the fog nodes for further +data analysis. We investigated the feasibility of our platform +by simulating the intended data flow of our design and the +effectiveness of our federated learning system. Lastly, our +implementation aims to provide a low-cost setup that simulates +our platform under a resource-constrained environment. We +presented a top-down view of our architecture and its data +flow. The following details will encompass the other design +choices we made for further establishing the predictive health- +care service that our design models. + +WearableloT +WearableloT +Device +Device +FogDevice +Cloud Server +WearableloT +WearableloT +Device +Device +FogDevice +FogDevice +WearableloT +WearableloT +Device +Device5 +B. Dataset and Neural Network +To establish the predictive healthcare service our platform +will use to test its feasibility, we looked for a dataset and +neural network configuration that fits our remote monitoring +data flow. We chose a standard classifier design and organized +the dataset to minimize its impact on our platform experi- +ments. Our focus is on the effectiveness of our platform in +securing wearable IoT device data and ensuring the integrity +and flexibility of the healthcare service, and not maximizing +the accuracy of the classifier. Therefore, the classifier we +used in testing our platform is for human activity recognition +(HAR). It uses accelerometer data from various commercial +wearable IoT devices such as smartphones and smartwatches +to determine the posture and condition of each user [35]. +Healthcare services use these to monitor the safety of their +patients. For example, it can detect the falling pattern of a +person and enable real-time response. Also, they can monitor +people’s daily exercise and regimen for further consultation +and improvements for their health [36]. Similarly, we will +use it to simulate patient data for training HAR models to +better reactive analysis and accident detection in wearable IoT +device-based remote monitoring. +The dataset we selected is the Human Activity Recogni- +tion Using Smartphones Data Set from the UCI Machine +Learning Repository [37]. It is from an experiment with 30 +volunteers within 19-48 years. Each individual performed +six activities: walking, walking upstairs, walking downstairs, +sitting, standing, and lying down. Each volunteer did each +action wearing a Samsung Galaxy SII smartphone on their +waist. The result is a dataset that contains 10299 instances +of 3-axial linear acceleration and 3-axial angular velocity +captured from the accelerometer and gyroscope of the mobile +device. It encompasses 561 labelled features with time and +frequency domain variables. We chose this classifier and +dataset combination because the data is from smartphones, +which is more common to the public. This selection provides +a better dataset scope that represents a larger population. Our +focus is on the federated learning aspect of the design. We +can minimize any impacts of extraneous variables that could +interfere with our experiments and the precision of its results +by using an already tested and documented classifier. Using it +provides convenience and efficiency as we simulate the HAR- +based predictive service testbed for our experiments. +We implement a one-dimension convolutional neural net- +work (CNN) because the dataset is a sequence with only a +single-dimensional source. Also, it is a design that minimizes +the overall processing requirement when training and testing +the model. Since we aim for a low-cost approach, we intend +to limit the impact of resource requirements. We coded it +using a combination of the Keras and Tensorflow libraries +in Python. The dataset was already pre-grouped to have a +70-30 split for training and testing. We use the same split +when training and testing our CNN to avoid generating any +significant impact from the nature of the classifier or the data. +Also, we refrained from potentially over-tuning the CNN by +only calibrating through the epoch number. We wanted to +ensure that the experiments focused on the performance of +the federated learning aspect of our design. +C. Federated Learning Implementation +We implemented the federated learning aspect to incorpo- +rate distributive learning with the HAR predictive model. We +used the previously presented CNN as the learning model for +our cloud and fog device. The target of the adopted CNN +is to learn from various human activities using accelerometer +and gyroscope data. The result is a HAR model for potential +anomaly detection in patient movement and activeness. Fed- +erated learning aims to solve for an optimal global model wG +that minimizes the weighted average loss of all clients. It takes +the global loss function L and simplifies it into a summation +of losses obtained from the local models wL of each client +shown in the following: +wG = arg min +wL +L(wL) = arg min +wL +� +1 +K +K +� +k=1 +Lk(wL) +� +, +(1) +where Li is the local loss function by client k. We translated +the solution for this federated learning problem as a python +script executed by the central server. It starts by iterating +through each local model we obtained through the fog devices. +Next, we extract the weights of each model and optimize +them by scaling them according to the total number of clients +and their accuracy. We optimized our weights by giving the +model with the highest accuracy the best scaling factor and +gave the rest lower scale values relative to the number of +total fog clients in the network. This scaling scheme ensures +that the best-performing local model is the base, and the rest +are additional references for further learning reinforcement. +While scaling, the server generates a CNN with the same layer +design. Then, the script aggregates the optimized weights via +averaging, takes the resulting solution, and sets it into the +prepared model, which becomes the global model. This model +encompasses all the knowledge obtained from each locally +trained CNN. A graphical representation that shows the flow +of our federated learning system is in Fig. 2. +D. Private Blockchain Implementation +We incorporated a private blockchain to manage the training +information shared by the devices that will train the HAR- +based predictive model. The blockchain protocols create an +access level ensuring that only trusted devices can access +the predictive model’s local training and global knowledge. +We coded it in Python. Each block is a class definition that +contains an ID, a list of training model results, and a hashed +string of the previous block. The private blockchain authorizes +communication between the server and the client based on +their assigned ID. Also, it collects the locally trained models +and stores them periodically within a block. These blocks +are chained cryptographically by generating a hash of the +previous block and storing it within the new one. The resulting +data structure is a log of every training model sent to the +server and the other actions the network carried out. The +blockchain becomes a hyperledger that stores the history of + +6 +Fig. 2: Flow of operations for our implemented federated +learning system within our platform. +every operation done by the network. Also, since it is tamper- +proof, these records are protected. We also initialize copies +of the blockchain and distribute them to each client to keep +the data decentralized and free from manipulation outside +the authorized devices. This distribution allows the service +to check the validity of each client and the local models they +are training and sending to the server. +With the private blockchain, the information and operation +within the network remain secure. Tampering becomes more +difficult with the hyperledger keeping track of all the changes +to the global model. It can record the results of each local +training through the obtained models from the fog clients. +Also, private and raw data from the clients are not sent to +the server, keeping them secure from any form of leakage or +malicious attacks. Overall, the blockchain sets up the network +by securing its endpoints and reinforcing its decentralization. +E. Hardware and Software Specifications +To simulate the predictive service based on the HAR model, +we selected hardware that fits our aim to have a low-cost +design. Our setup makes use of Raspberry Pi 3 Model Bs. +We chose Pis due to their portability, programmability, and +modularity for rapid programming. Also, we selected a low- +cost and low-end device to simulate the capabilities of most +wearable IoT devices. Each Pi has a Quad-core ARM Cortex +A53 processor that runs on a benchmark of 1.2 GHz. Our +focus is on model accuracy while processing speed is currently +not a concern. We also used Pis for both the fog devices and +the central server of our design. Each Pi is pre-loaded with a +Raspbian-Jesse operating system. We installed Python 3.6 on +every Pi as our programming language due to its flexibility and +diverse selection of open-source labels. The selected version is +the most recent version that works with all the Python libraries +needed for our design. As for the wearable IoT device that +represents the edge of our network, we based it on the same +smartphones from the HAR dataset we chose to use. It is a +Samsung Galaxy SII that uses a Dual-core 1.5 GHz Scorpion +processor. +IV. RESULTS AND DISCUSSIONS +We tested our proposed platform to evaluate its feasibility +by addressing data analysis security, integrity, and flexibility. +A. Testbed +We structured our testbed to simulate data flow from the +fog to the cloud server. It is to observe the behaviour of +our platform under the standard network interactions of local +and central servers of a fog-IoT network. In designing our +testbed, we use low-cost and low-end devices to evaluate our +design under minimal processing capabilities. We chose this +approach to show the feasibility of our proposal when using +simple IoT devices. It gives an estimate for a benchmark on +the minimum system requirements for our implementation. We +decided to use Raspberry Pi 3 Bs to model the fog devices +because it is modular, portable, and low-cost. During the early +stages of implementation, we discovered that the Pi could run +the classifier training and the blockchain scripts without any +issues. Therefore, we continued to use Pis as both the fog +and the cloud servers in our network. Another reason why +we chose to stay with just one type of technology was to +limit the potential impact of any processing or data capacity +advantage if we used a device with a better processor as a +server. Since Pi’s are modular, we replicated each node by +flashing an identical copy of the operating system to each +device. As a result, we arranged the network to have a Pi as +the cloud server with 10 Pis that serve as the fog nodes. +First, we assume that each fog server already contains the +wearable IoT device data, so the training and testing dataset +is pre-loaded in each Pi. Since our focus is on the federated +learning aspect carried out by the fog-IoT servers, we can skip +the simulations of data transmissions between the wearable +and the fog devices. Next, each fog device will train its models +locally and send them to the central cloud server via wireless +file transfer. The server will wait until it receives all the local +models before the next step. Next, it will aggregate the weights +of each model and scale it according to its accuracy. We +implemented a scaling scheme that gives the highest possible +scaling factor to the local model with the highest accuracy. At +the same time, the rest of the models will receive the same +scaling factor but are significantly lower. Then, the server will +take the resulting scaled weights and set them on a structurally +identical CNN, resulting in the global model. For our test, we +plan to compare the accuracy of the global model with the +average of the locally trained models from each fog device. +We elected to use this metric to evaluate the behaviour of + +Cloud Server +Global Model +CNN +Setting +Aggregated Weights +Local Model +Local Model +Local Model +CNN +CNN +CNN +Training) +Training +Training +Local Data +Local Data +Local Data +Fog Device +Fog Device +FogDevice +WearableloTdevices +WearableloTdevices +WearableloTdevices7 +Fig. 3: Sample training and validation loss of one local model +yielding a testing accuracy of 90.43%. +our platform. Also, we aim to ensure that our implemented +federated learning system does not disrupt the integrity of the +classifier. Finally, to investigate the security and adaptability +of our design, we later discuss the strengths of our platform +that we observed as we tested and simulated the platform. +B. Integrity Evaluation via Training Model Accuracy +Our test investigated the ability of our platform to maintain +the integrity of the predictive data analysis of the classifier +within the healthcare network. We used a 1D CNN to classify +HAR using the HAR dataset from the UCI Machine Learning +Repository. We chose to evaluate the feasibility of our design +by comparing the accuracy between the global and the local +models. +First, we trained the model locally using 10 Pis pre-loaded +with the classifier and the dataset. We arranged the models +by assigning a number to each corresponding client. The +fog devices trained each local model using the same 70% +split of the dataset. Also, we tested each model with the +remaining 30% of the dataset. We tried to minimize the +complexity of our classifier and focused on the performance +of our federated learning system by doing the minimal tuning. +The only parameters we tinkered with were the epoch number +and the batch size, which we set to 10 and 8, respectively. +The resulting model yielded a 90.43% testing accuracy. We +generated its validation-to-training loss and accuracy plots +as shown in Fig. 3 and Fig. 4, respectively. These results +are a sample of 1 out of 10 fog clients. Each presents +the reasonable validity of the classifier at minimal tuning +in correctly identifying the different labelled human actions +based on accelerometer and gyroscope data. +Upon further analyzing these graphs, we observed a spike in +the validation loss on epoch 6 of Fig. 3. This behaviour is due +to the learning rate causing the trained model to be volatile at +this point. Since the graph stabilizes, it does not significantly +impact the overall model. Also, there is a similar spike on +epoch 5 of Fig. 4. This behaviour could reflect potential +Fig. 4: Sample training and validation accuracy of one local +model yielding a testing accuracy of 90.43%. +Fig. 5: Confusion matrix of testing the global model with +aggregated weights from 10 local models. +volatility within the dataset. However, since the amplitude of +the fluctuations is within a reasonable range of +/- 2% from +the median, it shows that the data’s volatility is not significant +enough to render the model inaccurate. +Overall, each trained model yielded similar behavioural +trends but had varying final accuracy values within a range +of +/- 1.28% from the median. This range shows the impact +of the data on the accuracy of the model. However, it is not +significant enough to invalidate the training results. Therefore, +we took the average of their performances and used it to +represent the benchmark of the classifier for each collective +number of clients. After getting the average accuracy of +the local models, we compared it with the global model to +evaluate our implemented federated learning system. +First, we generated the global model for each N number +of clients. We have a visualization of our global model’s +performance after aggregating the weights of local models +from 10 clients through the confusion matrix in Fig. 5. Next, +we compared the average accuracy of the N local models +to their corresponding global models. This method allows + +0.8 +Training Loss +Validation Loss +0.7 +0.6- +0.5 +Loss +0.4 +0.3 +0.2 +0.1 +0 +2 +4 +6 +8 +Epoch0.96 +0.94 +Accuracy +0.92 +Training Accuracy +Validation Accuracy +0.90 +0.88 +0.86 +0.84 +0 +2 +4 +6 +8 +Epoch500 +walking +470 +3 +23 +0 +0 +0 +walk_upstairs +16 +429 +25 +1 +0 +0 +400 +Class +walk_downstairs +1 +4 +415 +0 +0 +0 +300 +Original +sitting +0 +24 +0 +408 +57 +2 +200 +standing +1 +1 +0 +60 +470 +0 +100 +lying +0 +27 +0 +0 +0 +510 +0 +ups +walk +Predicted Class8 +Fig. 6: Difference in testing accuracy between global model +and average local model as the number of clients increases. +us to see the model’s ability to keep up with the average +performance of a collection of locally trained models. We +obtained the average accuracy of the local models while +increasing the number of clients. Then, we generated their +corresponding global model and recorded its accuracy. +Finally, we plotted the comparison between the accuracy +of the global model and the averaged local model shown +in Fig. 6. Based on the results, we can observe that as the +number of clients increases, the accuracy of the global model +keeps up and even eventually performs better than the average +accuracy of its local models. Through this observation, we can +see the ability of our implemented federated learning system to +minimize any loss from aggregating weights. We can attribute +this improvement in accuracy to our optimization scheme that +assigns a higher scaling factor for the local model with the +best performance. Also, we can see how the accuracy of the +global model peaks at 91.75%. +We can conclude that our scaling scheme can maximize the +classifier’s performance by prioritizing the local models that +yield the best results. Also, the platform performed effectively +even with the low-cost devices functioning as servers. There- +fore, we can further reinforce the argument that our design +can be scalable and efficient even under resource constraints. +Since the Pis are modular and portable, our platform can cover +remote monitoring applications constrained by distance and +mobility. Overall, this shows the potential of our design as +an effective and scalable approach that can benefit wearable +IoT device-based predictive healthcare even with resource +constraints. +C. Security and Adaptability Analysis +The following is an analysis of the security and adaptability +of our design. Each evaluation is from observations from our +simulations and testing: +1) Security: Our design choices aim to preserve the data +privacy of wearable IoT devices. Federated learning allows +us to keep raw data from leaving the local network. Since +only the models are transmitted, limiting potential leakages +protects the data of wearable IoT devices. With blockchains, +our platform can create a more exclusive network where +only trusted devices can send data to the cloud server. As +a result, the hyperledger can reduce the endpoints that ex- +pose the server to malicious attacks such as spoofing and +impersonation. Also, this can reduce the threat of Denial of +Services (DoS) attacks. Since the blockchain can regulate +which devices can actively communicate with the server, it +can serve as a rate limiter for any targeted attack. Blockchains +provide an extra level of security due to their built-in access +control and cryptographic structure. Overall, each technology +integrated into our platform can preserve the data privacy of +wearable IoT devices. +2) Adaptability: Based on our observations, our fog-based +IoT architecture was able to aid in implementing both feder- +ated learning and blockchain technology with ease. We can +attribute it to the decentralized structure of fog-IoT. Due to +the fog architecture, both technologies can perform optimally +under this network arrangement and utilize their strengths. +Device diversity causes network strains due to the constant +demand for updates and maintenance. However, federated +learning reduces this strain by standardizing transmitted in- +formation into a uniform package. This package is the locally +trained model. Lessening the demand for the cloud server to +keep up with the latest wearable device allows it to focus on +deploying services efficiently while keeping up with the most +recent global knowledge. As long as the classifier structure +remains uniform, the server only needs to regenerate the same +model with different weights. As a result, the regeneration and +aggregation become modular to the service. This modularity +contributes to the adaptability of the network. It enables the +use of other classifiers and wearable IoT device groups. +Keeping the network decentralized through the fog also +allows better clustering. This design reduces the time required +to update all its servers and devices. For instance, take a +heterogeneity equation H based on the distribution of worker +update time φw presented in the following: +H = 1 − +1 +W − 1 +W −1 +� +w=1 +φW +φw +, +(2) +According to Eq. 2, a higher worker count W lowers the +overall heterogeneity H of the network. Also, we assume +that φW = Min(φ1...φw). This assumption means that the +heterogeneity value of the network is proportional to how +spread out is the update times of each worker. The higher this +value H, the lesser the network is impacted by the updates +times of its devices. In a standard cloud IoT network, the +number of workers equals the number of wearable IoT devices +it holds. As a result, the heterogeneity value will be high +due to the diversity of the wearable devices connecting to the +server. With varying update times, the �W −1 +w=1 +φW +φw term will +be asymptotic to 0. When plugged in the rest of the equation, +the heterogeneity value is maximized and asymptotic to +1. +The closer this value is to 1, the more heterogeneous the +network. Our fog-based IoT approach allows us to reduce the +impacts of the update time of each worker by standardizing +them. Instead of having each wearable IoT device as a separate + +91.8 +Global +Local (Averaged) +91.6 +91.4 +(%) +Accuracy +91.2 +91.0 +90.8 +90.6 +4 +9 +8 +10 +Numberof Clients9 +worker, they are clustered by the fog device and become the +new worker. With standardized devices, the �W −1 +w=1 +φW +φw term +will be asymptotic to W. When plugged in the rest of the +equation, the heterogeneity value is minimized and asymptotic +to -1. The closer this value is to -1, the less heterogeneous +the network. Ideally, we aim to minimize the heterogeneity +value to indicate a network less affected by the diversity of +its worker’s update times. +With our experiments, the update time is each fog device’s +training and communication time. We can observe in the +testbed design that having a distributive arrangement can +reduce the impact of the diversity of wearable IoT devices +on the overall update time by offloading the training and +processing to the fog. Instead of many wearable IoT devices +that can yield varying update times, sending training data to +the cloud server can be standardized. So theoretically, we can +infer that our platform will result in a lower heterogeneity +value. This analysis highlights the benefits of the distributive +approach that we proposed. +Also, the resulting accuracy of the global model presents +how the precision of the predictive healthcare service is +not affected by the shift in the location of the learning +process. Since the edge device is only required to send data +securely, it does not need standardization as long as the fog +device is aware of the data it receives. Also, having one less +complex process within a sensing device makes its design +more robust. Removing this constraint helps the server keep +up with the constant introduction of new wearable IoT devices +by simplifying the data flow within the network. This change +creates a more adaptable network that can cater to a diverse +pool of wearable IoT devices. +D. Future Work and Recommendations +Further improvements to the design can be using other +metrics for testing. Another performance metric we can +include for future iterations can be the propagation delay +between the fog and the cloud when transporting the training +model results. This addition introduces potential dynamic +scenarios that further test the mobility and reach of our +platform. Another metric could be the power consumption +of the servers. The coverage of servers must reach remote +areas in extreme cases of predictive healthcare. As a result, +more portable and power-efficient designs are in demand for +sustainability and longer service uptime. This design addition +introduces the potential design and device optimizations that +make the platform more cost-efficient while managing the +predictive healthcare service. Another improvement is to test +the platform with more than one neural network design and +dataset. This addition can further emphasize our platform’s +modularity and performance under different configurations. +V. CONCLUSION +We propose a platform that addresses the issues of data +privacy, service integrity, and network structure adaptability +of wearable IoT devices in predictive healthcare. We used +federated learning for its ability to effectively aggregate local +models into a global entity to ensure the integrity of the +predictive service. We further incorporated private blockchain +technology to reinforce the overall network security. Lastly, +we have fog-IoT as the base for offloading and process redis- +tribution. By evaluating the implemented federated learning +system in terms of model accuracy, we observed its feasibility +in maintaining the integrity of the HAR classifier. Next, we +discussed the effectiveness of our platform even when using +a low-cost and low-end device as its fog and cloud server. +Then, we analyzed our design choices and highlighted its +strengths in terms of privacy preservation, security, and net- +work adaptability. Overall, through our testing and evaluation, +we saw the feasibility and potential of our proposed platform +in addressing the security, integrity, and adaptive issues of +wearable IoT devices in predictive healthcare. +REFERENCES +[1] A. Hussain, K. Zafar, and A. 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Reyes-Ortiz, “A public +domain dataset for human activity recognition using smartphones,” 01 +2013. + diff --git a/VdE3T4oBgHgl3EQfbApG/content/tmp_files/load_file.txt b/VdE3T4oBgHgl3EQfbApG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..825a1fef5469d5269ebbd848b1793578a376301d --- /dev/null +++ b/VdE3T4oBgHgl3EQfbApG/content/tmp_files/load_file.txt @@ -0,0 +1,805 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf,len=804 +page_content='1 Federated Learning and Blockchain-enabled Fog-IoT Platform for Wearables in Predictive Healthcare Marc Jayson Baucas, Student Member, IEEE, Petros Spachos, Senior Member, IEEE, and Konstantinos N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Plataniotis, Fellow, IEEE Abstract—Over the years, the popularity and usage of wear- able Internet of Things (IoT) devices in several healthcare services are increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Among the services that benefit from the usage of such devices is predictive analysis, which can improve early diagnosis in e-health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' However, due to the limitations of wearable IoT devices, challenges in data privacy, service integrity, and network structure adaptability arose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' To address these concerns, we propose a platform using federated learning and private blockchain technology within a fog-IoT network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' These technologies have privacy-preserving features securing data within the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We utilized the fog-IoT network’s distributive structure to create an adaptive network for wearable IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We designed a testbed to examine the proposed platform’s ability to preserve the integrity of a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Accord- ing to experimental results, the introduced implementation can effectively preserve a patient’s privacy and a predictive service’s integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We further investigated the contributions of other technologies to the security and adaptability of the IoT network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Overall, we proved the feasibility of our platform in addressing significant security and privacy challenges of wearable IoT devices in predictive healthcare through analysis, simulation, and experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Index Terms—Machine learning, Data privacy, Predictive models, Distributed systems, Health care services, Platforms, Testbed, Health informatics, Fog network, Security, Private Blockchain, Privacy, Scalability, Internet of Things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' INTRODUCTION T HE usage of wearable Internet of Things (IoT) devices in healthcare is rising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Due to their availability and sens- ing capability, these devices collect physiological data from patients and provide real-time diagnosis [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Wearable IoT de- vices have caused remote healthcare to make a paradigm shift into predictive diagnosis and reliable early detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The data collected by these devices partnered with different learning techniques aid in predictive healthcare services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Doctors can analyze data such as their patient’s activities and accurately predict anomalies and threats against their health [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' They can also prescribe treatments for preventing and addressing these detected concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' However, this breakthrough has limitations This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Baucas and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Spachos are with the School of Engineering, University of Guelph, Guelph, ON, N1G2W1, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' (e-mail: baucas@uoguelph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='ca;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' petros@uoguelph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='ca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Plataniotis is with the Department of Electrical and Computer En- gineering, University of Toronto, Toronto, ON, M5S3G4, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' (e-mail: kostas@ece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='utoronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='ca) in its technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Challenge within a network that employs wearable IoT devices cause impasses in predictive healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' An open issue for wearable IoT devices in predictive healthcare is the amount of data it needs to be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' A large amount of personal data is collected, resulting in security and privacy concerns due to the nature of the data used for analysis [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The wearable devices’ limitations on processing capabilities lead to vulnerabilities and potential leakages in sensitive patient information [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Another issue is the integrity and reliability of the service [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Structuring healthcare to prioritize certain aspects can cause trade-offs in others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Service integrity is crucial for this field in remote healthcare that relies on wearable data accuracy and predictive model precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' One more issue is the adaptability of the network that deploys and serves these predictive healthcare services [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Wearable IoT device standardization is a significant concern to IoT networks due to the heterogeneity it introduces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This diversity results in demands for continuous maintenance and updates to the medical server to ensure that it is up to date with every newly introduced wearable IoT device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As a result, concerns about adaptability limit the healthcare network from fully remaining relevant and sustainable over long stretches of its service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' In this work, we propose a fog-IoT platform to address these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We use federated learning to preserve patient data privacy and the integrity of the network’s predictive services [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, we incorporate blockchain technology to address the security issues in wearable IoT devices through its access control and cryptographic structure [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Finally, we combine these technologies within a fog-based IoT architec- ture to enforce decentralized servers and resource reallocation to improve the adaptability and sustainability of the overall network [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The main contributions of this work are: We present a fog-based IoT platform using federated learning and blockchain technology to preserve patient data privacy and improve the security of data within the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We designed a testbed that simulates and evaluates the proposed implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We used model accuracy to observe the platform’s ability to preserve the integrity of a predictive service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' A discussion on the background of our study and a brief literature review are in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Our proposed design and the methodology arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='04511v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='LG] 11 Jan 2023 2 followed are in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The presentation of a developed testbed and a discussion of the results are in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Finally, our conclusions are in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' BACKGROUND AND RELATED WORKS We provide a brief discussion on the relevant works in blockchain technology and federated learning implementa- tions for wearable IoT devices in healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Benefits of Using Wearable IoT Devices in Healthcare Wearable IoT devices can form a network of sensing devices that collect data from points of interest for predic- tive analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' In predictive healthcare, these devices collect physiological information from patients for better clinical decisions and to provide a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Each obtained statistic reflects the status of their health, which can identify future issues and potential risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' A benefit of using wearable IoT devices in healthcare is to reduce hospital congestion and medical examination costs via remote diagnoses and long- distance patient monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' In [11], they introduce a wearable Tele-ECG system for heart rate monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The proposed design merges the latest technologies in textile electrodes (TE), Bluetooth low energy (BLE), and smartphones to create a portable means of monitoring and evaluating the condition of a patient’s heart for potential anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' With features such as geographical tracking, medical history storage, and remote patient monitoring, the system establishes a light and cost- effective alternative for patients suffering from heart issues that need constant observation and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Another advantage of using wearable IoT devices in pre- dictive healthcare is improving the quality of early disease and fault detection in medical centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' In [12], they present a review of various biomedical IoT devices that raise the quality of diagnosis in healthcare services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The survey includes wear- able sensors as one of the leading technologies that enable advancements in predictive healthcare IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Services are now made remote and analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Patients can use technologies and wearable sensors that can help give doctors early information on potential causes and triggers of health complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As a result, there is improvement in predictive anomaly detection without constraints in geographical location and data resource availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, these devices widen the scope of medical centres toward their patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Doctors can use trend and behaviour analysis to pinpoint anomalies in a patient’s health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' At the same time, medical professionals can evaluate diseases with improved precision due to leveraged real-time and early detection [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Challenges of Using Wearable IoT Devices in Healthcare Although the usage of wearable IoT devices shows potential to raise the quality of services in healthcare, they also pose the following concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 1) Network Security and Data Privacy: Introducing wear- able devices to collect patient data adds more endpoints to the server of the healthcare service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Data collection and mon- itoring are convenient due to these remote services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' However, introducing more devices introduces more vulnerabilities to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Increasing the endpoints increases the potential areas where malicious users can attack and steal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As a result, there is a greater demand for better security as the network grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, privacy becomes a concern due to the sensitive information transmitted from the wearable device to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Therefore, network security and data privacy concerns grow as the network expands with more wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Different strategies and technologies have been proposed to address this issue [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' In [14], they present a multi- keyword search mechanism to preserve data privacy within the IoT network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' They aim to strengthen the encryption of the information transmitted from the patient to the medical centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This strategy achieves forward privacy preservation, which results in a more guaranteed security system for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Another example that aims for privacy preservation is in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' They use deep learning to secure the network by separating private from raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This strategy results in minimizing the chances of leaking sensitive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Our approach takes key strengths from these two strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We take the ability of cryptographic techniques to reinforce data transmissions and combine them with the adaptability of machine learning techniques to create a secure means of preserving patient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As a result, we chose blockchain technology for its strong encryption and federated learning for its adaptive ability to improve data privacy while keeping its trained models optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 2) Data Integrity and Precision: An advantage of using wearable IoT devices in healthcare is the real-time diagnosis and early detection of illness and medical anomalies within a patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' However, this can be affected by the quality of the sensor and the data processing scheme behind the service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Potential misdiagnosis is high if the sensor or the evaluation method is not up to standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Another source of error could come from how the data is collected and stored on the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Preserving physiological information integrity obtained from the wearable devices is needed as it travels from patient to server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It ensures that the data is recent and precise to the current status of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, the data handling during aggregation should be carried out with great precision so that the evaluation of the patient is accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' A service that can not collect, transmit, and process the data correctly and efficiently will result in inaccurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Therefore, the integrity and precision of information a concern that needs addressing for a healthcare service that uses wearable IoT devices to be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Approaches that improve the integrity of sensed data from IoT devices in healthcare services are proposed in [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' In [16], a novel data aggregator for efficient and secure analysis in IoT monitoring is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Its scheme involves a cryptographic accumulator that securely collects data from wearable IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Another example is in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This work highlights the advantages of integrating blockchains with IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' They aim to enhance the performance of healthcare services that use wearable IoT devices by using the security advantages of blockchain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Similar to the approach in [16], both strategies focus on the ability of cryptographic technologies 3 to ensure the accuracy of sensed data within the IoT network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Our proposed approach differs as it incorporates federated learning for a more secure data aggregator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We combine it with decentralized organizations provided by blockchain and create a platform that ensures a secure medium for accurate data analysis in healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 3) Network Structure Adaptability and Flexibility: Wear- able IoT devices introduce different sensors and technologies that collect and process data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Due to their multiple advantages, healthcare services should include them in their systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' How- ever, innovations and improvements are iterative as changes arise often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Although this is a sign of technological growth and advancements in the healthcare sector, it dictates the pace at which current infrastructures need to keep up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Device diversity has always been a concern for IoT networks due to the need for ongoing standardization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As this new technology is still in development, services that use it should be able to follow the development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It is the same for wearable IoT devices used in healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The demand for technologies that detect each one is high as new diseases and virus strains often appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' However, every new technology introduced a need for the service to adapt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Networks that are less flexible result in losing their resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Some even end up losing their service and infrastructure altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Therefore, there is a need for adaptability and flexibility in the structure of healthcare networks for the continuously evolving nature of wearable IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Some approaches focus on improving the flexibility of wearable devices to tackle this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' In [18], they present an innovative wrist-worn prototype for patient monitoring, which also functions as an IoT gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Some sensors in remote healthcare focus on sensing the environment around the patient to ensure that the condition of their surrounding is beneficial to their health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Through it, ambient sensing devices can connect to the healthcare network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As a result, it sets up a platform that can easily integrate and synchronize other devices under it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' In [19], they present a wearable patch that functions similarly to the prototype in [18] by creating a flexible IoT gateway for connecting wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Instead of ambient sensors, their portal focuses on ECGs and PPGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Our approach is different as we aim to standardize devices through the fog by moving analysis closer to the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As a result, we focus on data management standardization instead of device standardization due to the diversity of wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Instead of specialization through prototypes, we plan to use fog-IoT paired with decentralized technologies such as blockchain and federated learning to develop a modular IoT network for wearable devices in healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Federated Learning for Wearable IoT Devices Federated learning is a machine learning technique that takes a distributed approach in training its models [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It uses its decentralized strategy to utilize global knowledge collected from its clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' In IoT, federated learning has two compo- nents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' the client and the server [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Its flow of operations for federated learning starts with each client training their model using their raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Then, the server aggregates and compiles the resulting models into a global model that it redistributes to each client’s use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As a result, each client receives the most optimal model given global knowledge gathered through local training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This procedure is ongoing, and the global model is updated every time the client provides new knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Federated learning is well-known for its capability of effec- tively preserving the privacy of data [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Requiring clients to transmit raw data directly to the server before it is analyzed can cause vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Without a strongly reinforced net- work, malicious attackers aiming to steal or tamper can target the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Federated learning protects raw data by creating a strategy that moves the analysis locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' There are lesser avenues for privacy leakages since the server only cares about the resulting trained model from each client [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As a result, this structure has an improved trusted architecture compared to standard network arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Aside from its security advantages, federated learning also improves the sustainability of IoT networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It provides a dynamic learning strategy that keeps global knowledge for services up to date [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Commonly, the server aggregates all the data first before it uses it to train the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Federated learning allows new correlations and behaviour changes from collected data to be detected by the server earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' A fully centralized scheme is slowed down and saturated due to the volume of the data it uses to train its models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Federated learning can reduce training time by running it locally and in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, there is a significant reduction in the data size as each client trains its model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Healthcare services that involve wearable IoT devices re- volve around continuous data sensing, learning, and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Federated learning can provide a sustainable and decentralized scheme to optimize these services in healthcare [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We chose to use this technology in our platform due to all the improvements it can contribute to IoT services, specifically healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Our proposed design is different because we aim to reinforce federated learning with the security provided by blockchains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, the foundation of our network will be a fog IoT network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The diversity of wearable IoT devices and their limited processing capabilities is a constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' However, we can overcome this by moving the training to the fog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This arrangement still allows training locally while considering the processing limits of wearable IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The result is a decentralized IoT network that can maximize the potential of federated learning in keeping data analysis in healthcare services secure and sustainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Blockchain for Wearable IoT Devices A blockchain is a cryptographic ledger that provides a dis- tributed service for information storage and security [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This decentralized control establishes a trusted system that will only grant access to users acknowledged by the blockchain ledger [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This feature protects the data from external or unwanted modifications and makes it immutable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' For wearable IoT devices in healthcare, the network can use blockchains as a tamper-proof database for patient data storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Its unique structure addresses different security issues in IoT networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' In [28], they use the unique consent mechanism 4 of blockchains to secure user data privacy in wearable fitness devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' They aim to prove the feasibility of the trustwor- thiness of blockchains when fortifying the data privacy of wearable IoT device data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Another proposed use for the access control of blockchains is in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' They developed a lightweight authentication scheme that classifies mobile devices and dif- ferentiates them from fake data injections or illegitimate users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Their designs show the potential of blockchain technology in securing the healthcare server and the monitoring devices used by its services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' In [30], they use a different feature of blockchains in reinforcing their healthcare service that uses wearable IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' They integrate the encryption model of blockchains to improve the security of the IoT network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Their design creates a searchable encryption technique that assists in securing collected COVID-19 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It shows the effectiveness of blockchain architecture in helping protect the healthcare IoT network from several security threats like malicious data injection and hijacking sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Our design differs from the previously presented imple- mentations because we aim to use a private blockchain [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Public blockchains use a reward-based protocol called “proof- of-work” (PoW) in granting access to their devices [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Although this protocol is secure, its processing requirement is a caveat for wearable IoT devices in healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' In [33], we highlighted how some wearable IoT devices are incapable of adapting to this authentication structure due to their low- cost and low-end designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Therefore, we need a different blockchain architecture that can cater to the design nature of wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Private blockchains function differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' They incorporate a more trust-based protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This approach turns the blockchain into a hyperledger that can regulate its devices based on a list of trusted devices defined by the network owner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Healthcare IoT networks can benefit from this approach more since the need for high processing power for their wearable devices is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As a result, the types of IoT-based wearable devices they can incorporate into their services do not limit the healthcare network [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' DESIGN AND METHODOLOGY The following is a discussion of our proposed platform, including a description of the design and the different com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Architecture Overview We propose a fog-IoT-based approach to secure data ex- change from wearable IoT devices used for predictive health- care services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We aim to improve the overall structure’s service integrity and flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' To further enhance the distributive organization of our network, we chose to use federated learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' With its ability to aggregate learning models, we can sort our data and keep private information secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, it can reduce the sources of leakage since each client is only required to send their locally trained model to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Integrating federated learning improves the sustainability of the predictive healthcare service by providing a system that can organize the model data and provide a global model that represents the overall knowledge obtained through all the local training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 1: Network arrangement for the proposed fog-based platform for wearable IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The result is a scalable design that introduces flexibility by keeping the network distributed while ensuring that it does not impede the learning and analysis of the service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Next, we combine federated learning with private blockchain technology for more secure client authorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, it is treated as a hyperledger to store the IDs of each client device and locally trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Due to its immutability, tracking and logging each trained model for version control and debugging improves the data analysis and learning aspect of healthcare networks and their services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Commercial wearable devices are diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Some are low- end by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' To address this, we moved the process of locally training the model from the edge of the network to the fog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We can reallocate the training process by providing an intermediary fog device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Usually, training takes a large portion of processor capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Unfortunately, not every wearable IoT device can do it effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' For our platform to be able and handle a diverse pool of wearable devices, we offloaded the processing required to a fog device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The proposed network arrangement is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' First, the fog device collects all sensed data from each wearable IoT device and trains the model locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Each one sends its local model to the server for storage and tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Next, the server aggregates all the knowledge from the local models and generates a global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Then, the server sends this model to each wearable IoT device through the fog nodes for further data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We investigated the feasibility of our platform by simulating the intended data flow of our design and the effectiveness of our federated learning system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Lastly, our implementation aims to provide a low-cost setup that simulates our platform under a resource-constrained environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We presented a top-down view of our architecture and its data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The following details will encompass the other design choices we made for further establishing the predictive health- care service that our design models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' WearableloT WearableloT Device Device FogDevice Cloud Server WearableloT WearableloT Device Device FogDevice FogDevice WearableloT WearableloT Device Device5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Dataset and Neural Network To establish the predictive healthcare service our platform will use to test its feasibility, we looked for a dataset and neural network configuration that fits our remote monitoring data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We chose a standard classifier design and organized the dataset to minimize its impact on our platform experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Our focus is on the effectiveness of our platform in securing wearable IoT device data and ensuring the integrity and flexibility of the healthcare service, and not maximizing the accuracy of the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Therefore, the classifier we used in testing our platform is for human activity recognition (HAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It uses accelerometer data from various commercial wearable IoT devices such as smartphones and smartwatches to determine the posture and condition of each user [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Healthcare services use these to monitor the safety of their patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' For example, it can detect the falling pattern of a person and enable real-time response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, they can monitor people’s daily exercise and regimen for further consultation and improvements for their health [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Similarly, we will use it to simulate patient data for training HAR models to better reactive analysis and accident detection in wearable IoT device-based remote monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The dataset we selected is the Human Activity Recogni- tion Using Smartphones Data Set from the UCI Machine Learning Repository [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It is from an experiment with 30 volunteers within 19-48 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Each individual performed six activities: walking, walking upstairs, walking downstairs, sitting, standing, and lying down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Each volunteer did each action wearing a Samsung Galaxy SII smartphone on their waist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The result is a dataset that contains 10299 instances of 3-axial linear acceleration and 3-axial angular velocity captured from the accelerometer and gyroscope of the mobile device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It encompasses 561 labelled features with time and frequency domain variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We chose this classifier and dataset combination because the data is from smartphones, which is more common to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This selection provides a better dataset scope that represents a larger population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Our focus is on the federated learning aspect of the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We can minimize any impacts of extraneous variables that could interfere with our experiments and the precision of its results by using an already tested and documented classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Using it provides convenience and efficiency as we simulate the HAR- based predictive service testbed for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We implement a one-dimension convolutional neural net- work (CNN) because the dataset is a sequence with only a single-dimensional source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, it is a design that minimizes the overall processing requirement when training and testing the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Since we aim for a low-cost approach, we intend to limit the impact of resource requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We coded it using a combination of the Keras and Tensorflow libraries in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The dataset was already pre-grouped to have a 70-30 split for training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We use the same split when training and testing our CNN to avoid generating any significant impact from the nature of the classifier or the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, we refrained from potentially over-tuning the CNN by only calibrating through the epoch number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We wanted to ensure that the experiments focused on the performance of the federated learning aspect of our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Federated Learning Implementation We implemented the federated learning aspect to incorpo- rate distributive learning with the HAR predictive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We used the previously presented CNN as the learning model for our cloud and fog device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The target of the adopted CNN is to learn from various human activities using accelerometer and gyroscope data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The result is a HAR model for potential anomaly detection in patient movement and activeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Fed- erated learning aims to solve for an optimal global model wG that minimizes the weighted average loss of all clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It takes the global loss function L and simplifies it into a summation of losses obtained from the local models wL of each client shown in the following: wG = arg min wL L(wL) = arg min wL � 1 K K � k=1 Lk(wL) � , (1) where Li is the local loss function by client k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We translated the solution for this federated learning problem as a python script executed by the central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It starts by iterating through each local model we obtained through the fog devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Next, we extract the weights of each model and optimize them by scaling them according to the total number of clients and their accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We optimized our weights by giving the model with the highest accuracy the best scaling factor and gave the rest lower scale values relative to the number of total fog clients in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This scaling scheme ensures that the best-performing local model is the base, and the rest are additional references for further learning reinforcement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' While scaling, the server generates a CNN with the same layer design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Then, the script aggregates the optimized weights via averaging, takes the resulting solution, and sets it into the prepared model, which becomes the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This model encompasses all the knowledge obtained from each locally trained CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' A graphical representation that shows the flow of our federated learning system is in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Private Blockchain Implementation We incorporated a private blockchain to manage the training information shared by the devices that will train the HAR- based predictive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The blockchain protocols create an access level ensuring that only trusted devices can access the predictive model’s local training and global knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We coded it in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Each block is a class definition that contains an ID, a list of training model results, and a hashed string of the previous block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The private blockchain authorizes communication between the server and the client based on their assigned ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, it collects the locally trained models and stores them periodically within a block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' These blocks are chained cryptographically by generating a hash of the previous block and storing it within the new one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The resulting data structure is a log of every training model sent to the server and the other actions the network carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The blockchain becomes a hyperledger that stores the history of 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 2: Flow of operations for our implemented federated learning system within our platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' every operation done by the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, since it is tamper- proof, these records are protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We also initialize copies of the blockchain and distribute them to each client to keep the data decentralized and free from manipulation outside the authorized devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This distribution allows the service to check the validity of each client and the local models they are training and sending to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' With the private blockchain, the information and operation within the network remain secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Tampering becomes more difficult with the hyperledger keeping track of all the changes to the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It can record the results of each local training through the obtained models from the fog clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, private and raw data from the clients are not sent to the server, keeping them secure from any form of leakage or malicious attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Overall, the blockchain sets up the network by securing its endpoints and reinforcing its decentralization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Hardware and Software Specifications To simulate the predictive service based on the HAR model, we selected hardware that fits our aim to have a low-cost design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Our setup makes use of Raspberry Pi 3 Model Bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We chose Pis due to their portability, programmability, and modularity for rapid programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, we selected a low- cost and low-end device to simulate the capabilities of most wearable IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Each Pi has a Quad-core ARM Cortex A53 processor that runs on a benchmark of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Our focus is on model accuracy while processing speed is currently not a concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We also used Pis for both the fog devices and the central server of our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Each Pi is pre-loaded with a Raspbian-Jesse operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We installed Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='6 on every Pi as our programming language due to its flexibility and diverse selection of open-source labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The selected version is the most recent version that works with all the Python libraries needed for our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As for the wearable IoT device that represents the edge of our network, we based it on the same smartphones from the HAR dataset we chose to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It is a Samsung Galaxy SII that uses a Dual-core 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='5 GHz Scorpion processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS We tested our proposed platform to evaluate its feasibility by addressing data analysis security, integrity, and flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Testbed We structured our testbed to simulate data flow from the fog to the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It is to observe the behaviour of our platform under the standard network interactions of local and central servers of a fog-IoT network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' In designing our testbed, we use low-cost and low-end devices to evaluate our design under minimal processing capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We chose this approach to show the feasibility of our proposal when using simple IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It gives an estimate for a benchmark on the minimum system requirements for our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We decided to use Raspberry Pi 3 Bs to model the fog devices because it is modular, portable, and low-cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' During the early stages of implementation, we discovered that the Pi could run the classifier training and the blockchain scripts without any issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Therefore, we continued to use Pis as both the fog and the cloud servers in our network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Another reason why we chose to stay with just one type of technology was to limit the potential impact of any processing or data capacity advantage if we used a device with a better processor as a server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Since Pi’s are modular, we replicated each node by flashing an identical copy of the operating system to each device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As a result, we arranged the network to have a Pi as the cloud server with 10 Pis that serve as the fog nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' First, we assume that each fog server already contains the wearable IoT device data, so the training and testing dataset is pre-loaded in each Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Since our focus is on the federated learning aspect carried out by the fog-IoT servers, we can skip the simulations of data transmissions between the wearable and the fog devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Next, each fog device will train its models locally and send them to the central cloud server via wireless file transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The server will wait until it receives all the local models before the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Next, it will aggregate the weights of each model and scale it according to its accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We implemented a scaling scheme that gives the highest possible scaling factor to the local model with the highest accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' At the same time, the rest of the models will receive the same scaling factor but are significantly lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Then, the server will take the resulting scaled weights and set them on a structurally identical CNN, resulting in the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' For our test, we plan to compare the accuracy of the global model with the average of the locally trained models from each fog device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We elected to use this metric to evaluate the behaviour of Cloud Server Global Model CNN Setting Aggregated Weights Local Model Local Model Local Model CNN CNN CNN Training) Training Training Local Data Local Data Local Data Fog Device Fog Device FogDevice WearableloTdevices WearableloTdevices WearableloTdevices7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 3: Sample training and validation loss of one local model yielding a testing accuracy of 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='43%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' our platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, we aim to ensure that our implemented federated learning system does not disrupt the integrity of the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Finally, to investigate the security and adaptability of our design, we later discuss the strengths of our platform that we observed as we tested and simulated the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Integrity Evaluation via Training Model Accuracy Our test investigated the ability of our platform to maintain the integrity of the predictive data analysis of the classifier within the healthcare network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We used a 1D CNN to classify HAR using the HAR dataset from the UCI Machine Learning Repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We chose to evaluate the feasibility of our design by comparing the accuracy between the global and the local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' First, we trained the model locally using 10 Pis pre-loaded with the classifier and the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We arranged the models by assigning a number to each corresponding client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The fog devices trained each local model using the same 70% split of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, we tested each model with the remaining 30% of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We tried to minimize the complexity of our classifier and focused on the performance of our federated learning system by doing the minimal tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The only parameters we tinkered with were the epoch number and the batch size, which we set to 10 and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The resulting model yielded a 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='43% testing accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We generated its validation-to-training loss and accuracy plots as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' These results are a sample of 1 out of 10 fog clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Each presents the reasonable validity of the classifier at minimal tuning in correctly identifying the different labelled human actions based on accelerometer and gyroscope data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Upon further analyzing these graphs, we observed a spike in the validation loss on epoch 6 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This behaviour is due to the learning rate causing the trained model to be volatile at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Since the graph stabilizes, it does not significantly impact the overall model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, there is a similar spike on epoch 5 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This behaviour could reflect potential Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 4: Sample training and validation accuracy of one local model yielding a testing accuracy of 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='43%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 5: Confusion matrix of testing the global model with aggregated weights from 10 local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' volatility within the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' However, since the amplitude of the fluctuations is within a reasonable range of +/- 2% from the median, it shows that the data’s volatility is not significant enough to render the model inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Overall, each trained model yielded similar behavioural trends but had varying final accuracy values within a range of +/- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='28% from the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This range shows the impact of the data on the accuracy of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' However, it is not significant enough to invalidate the training results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Therefore, we took the average of their performances and used it to represent the benchmark of the classifier for each collective number of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' After getting the average accuracy of the local models, we compared it with the global model to evaluate our implemented federated learning system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' First, we generated the global model for each N number of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We have a visualization of our global model’s performance after aggregating the weights of local models from 10 clients through the confusion matrix in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Next, we compared the average accuracy of the N local models to their corresponding global models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This method allows 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='8 Training Loss Validation Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='6- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='5 Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='1 0 2 4 6 8 Epoch0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='94 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='92 Training Accuracy Validation Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='84 0 2 4 6 8 Epoch500 walking 470 3 23 0 0 0 walk_upstairs 16 429 25 1 0 0 400 Class walk_downstairs 1 4 415 0 0 0 300 Original sitting 0 24 0 408 57 2 200 standing 1 1 0 60 470 0 100 lying 0 27 0 0 0 510 0 ups walk Predicted Class8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 6: Difference in testing accuracy between global model and average local model as the number of clients increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' us to see the model’s ability to keep up with the average performance of a collection of locally trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We obtained the average accuracy of the local models while increasing the number of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Then, we generated their corresponding global model and recorded its accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Finally, we plotted the comparison between the accuracy of the global model and the averaged local model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Based on the results, we can observe that as the number of clients increases, the accuracy of the global model keeps up and even eventually performs better than the average accuracy of its local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Through this observation, we can see the ability of our implemented federated learning system to minimize any loss from aggregating weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We can attribute this improvement in accuracy to our optimization scheme that assigns a higher scaling factor for the local model with the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, we can see how the accuracy of the global model peaks at 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We can conclude that our scaling scheme can maximize the classifier’s performance by prioritizing the local models that yield the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, the platform performed effectively even with the low-cost devices functioning as servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' There- fore, we can further reinforce the argument that our design can be scalable and efficient even under resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Since the Pis are modular and portable, our platform can cover remote monitoring applications constrained by distance and mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Overall, this shows the potential of our design as an effective and scalable approach that can benefit wearable IoT device-based predictive healthcare even with resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Security and Adaptability Analysis The following is an analysis of the security and adaptability of our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Each evaluation is from observations from our simulations and testing: 1) Security: Our design choices aim to preserve the data privacy of wearable IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Federated learning allows us to keep raw data from leaving the local network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Since only the models are transmitted, limiting potential leakages protects the data of wearable IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' With blockchains, our platform can create a more exclusive network where only trusted devices can send data to the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As a result, the hyperledger can reduce the endpoints that ex- pose the server to malicious attacks such as spoofing and impersonation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, this can reduce the threat of Denial of Services (DoS) attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Since the blockchain can regulate which devices can actively communicate with the server, it can serve as a rate limiter for any targeted attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Blockchains provide an extra level of security due to their built-in access control and cryptographic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Overall, each technology integrated into our platform can preserve the data privacy of wearable IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 2) Adaptability: Based on our observations, our fog-based IoT architecture was able to aid in implementing both feder- ated learning and blockchain technology with ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We can attribute it to the decentralized structure of fog-IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Due to the fog architecture, both technologies can perform optimally under this network arrangement and utilize their strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Device diversity causes network strains due to the constant demand for updates and maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' However, federated learning reduces this strain by standardizing transmitted in- formation into a uniform package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This package is the locally trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Lessening the demand for the cloud server to keep up with the latest wearable device allows it to focus on deploying services efficiently while keeping up with the most recent global knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As long as the classifier structure remains uniform, the server only needs to regenerate the same model with different weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As a result, the regeneration and aggregation become modular to the service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This modularity contributes to the adaptability of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' It enables the use of other classifiers and wearable IoT device groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Keeping the network decentralized through the fog also allows better clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This design reduces the time required to update all its servers and devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' For instance, take a heterogeneity equation H based on the distribution of worker update time φw presented in the following: H = 1 − 1 W − 1 W −1 � w=1 φW φw , (2) According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' 2, a higher worker count W lowers the overall heterogeneity H of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, we assume that φW = Min(φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='φw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This assumption means that the heterogeneity value of the network is proportional to how spread out is the update times of each worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The higher this value H, the lesser the network is impacted by the updates times of its devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' In a standard cloud IoT network, the number of workers equals the number of wearable IoT devices it holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As a result, the heterogeneity value will be high due to the diversity of the wearable devices connecting to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' With varying update times, the �W −1 w=1 φW φw term will be asymptotic to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' When plugged in the rest of the equation, the heterogeneity value is maximized and asymptotic to +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The closer this value is to 1, the more heterogeneous the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Our fog-based IoT approach allows us to reduce the impacts of the update time of each worker by standardizing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Instead of having each wearable IoT device as a separate 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='8 Global Local (Averaged) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='6 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='4 (%) Accuracy 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='8 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content='6 4 9 8 10 Numberof Clients9 worker, they are clustered by the fog device and become the new worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' With standardized devices, the �W −1 w=1 φW φw term will be asymptotic to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' When plugged in the rest of the equation, the heterogeneity value is minimized and asymptotic to -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The closer this value is to -1, the less heterogeneous the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Ideally, we aim to minimize the heterogeneity value to indicate a network less affected by the diversity of its worker’s update times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' With our experiments, the update time is each fog device’s training and communication time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We can observe in the testbed design that having a distributive arrangement can reduce the impact of the diversity of wearable IoT devices on the overall update time by offloading the training and processing to the fog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Instead of many wearable IoT devices that can yield varying update times, sending training data to the cloud server can be standardized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' So theoretically, we can infer that our platform will result in a lower heterogeneity value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This analysis highlights the benefits of the distributive approach that we proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, the resulting accuracy of the global model presents how the precision of the predictive healthcare service is not affected by the shift in the location of the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Since the edge device is only required to send data securely, it does not need standardization as long as the fog device is aware of the data it receives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Also, having one less complex process within a sensing device makes its design more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Removing this constraint helps the server keep up with the constant introduction of new wearable IoT devices by simplifying the data flow within the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This change creates a more adaptable network that can cater to a diverse pool of wearable IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Future Work and Recommendations Further improvements to the design can be using other metrics for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Another performance metric we can include for future iterations can be the propagation delay between the fog and the cloud when transporting the training model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This addition introduces potential dynamic scenarios that further test the mobility and reach of our platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Another metric could be the power consumption of the servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' The coverage of servers must reach remote areas in extreme cases of predictive healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' As a result, more portable and power-efficient designs are in demand for sustainability and longer service uptime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This design addition introduces the potential design and device optimizations that make the platform more cost-efficient while managing the predictive healthcare service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Another improvement is to test the platform with more than one neural network design and dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' This addition can further emphasize our platform’s modularity and performance under different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' CONCLUSION We propose a platform that addresses the issues of data privacy, service integrity, and network structure adaptability of wearable IoT devices in predictive healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We used federated learning for its ability to effectively aggregate local models into a global entity to ensure the integrity of the predictive service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' We further incorporated private blockchain technology to reinforce the overall network security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Lastly, we have fog-IoT as the base for offloading and process redis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' By evaluating the implemented federated learning system in terms of model accuracy, we observed its feasibility in maintaining the integrity of the HAR classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Next, we discussed the effectiveness of our platform even when using a low-cost and low-end device as its fog and cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} +page_content=' Then, we analyzed our design choices and highlighted its strengths in terms of 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public domain dataset for human activity recognition using smartphones,” 01 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE3T4oBgHgl3EQfbApG/content/2301.04511v1.pdf'} diff --git a/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf b/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d7a00d771640923d67c54e4349c1f4cb020ff1a1 --- /dev/null +++ b/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4fefe554742060bbf52aa03e6b1b3d41c0a295fb6622bbe843e5f57eaad30789 +size 320690 diff --git a/VdFIT4oBgHgl3EQfgyug/vector_store/index.pkl b/VdFIT4oBgHgl3EQfgyug/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..15bc8634e2590c5769a55a8906f8e3581216d1b0 --- /dev/null +++ b/VdFIT4oBgHgl3EQfgyug/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c33f4e4efdacebcb95cf05ab9fb58eda01469b7f9fc0d1c06e34790fca128d55 +size 201689 diff --git a/WNFJT4oBgHgl3EQf4S11/content/tmp_files/2301.11665v1.pdf.txt b/WNFJT4oBgHgl3EQf4S11/content/tmp_files/2301.11665v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..27c568adf7b150d861ffb8899594f717ca9dce2d --- /dev/null +++ b/WNFJT4oBgHgl3EQf4S11/content/tmp_files/2301.11665v1.pdf.txt @@ -0,0 +1,754 @@ +Hairy extension of the Bertotti-Robinson swallows almost everything +Vahideh Memari∗ and S. Habib Mazharimousavi† +Department of Physics, Faculty of Arts and Sciences, +Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 10, Turkey +(Dated: January 30, 2023) +A hairy extension of the Bertotti-Robinson regular spacetime has been recently introduced in +the context of the Einstein-Maxwell-Scaler theory that surprisingly is a closed singular black hole +[CQG39(2022)167001]. We investigate the geodesics of the null and timelike particles in this space- +time. We show that in the radial motion on the equatorial plane, while photons may collapse to the +singularity or escape to the edge of the universe a massive particle always collapses to the singular- +ity. Also, it is proven that on the equatorial plane there is no stable orbit not only for photons but +also for massive particles. However, the general geodesics of null and massive particles reveal that +all particles except the outgoing light ray, eventually fall into the black hole. This unique feature +makes closed black holes interesting for further studies. +I. +INTRODUCTION +In general relativity, no-hair theorems state that black holes are described by only three parameters, their mass +M, their electromagnetic charge Q, and their angular momentum ℓ [1–5]. Recent studies on black holes indicate the +existence of hairy black holes some of which form in the presence of Yang-Mills fields [6–9]. Hairy black holes are +solutions to Einstein’s equations in the interaction of diverse kinds of matter fields with gravity [10]. For instance +gravity in interaction with electromagnetism and axion [11–13], gravity coupled to electromagnetism and dilaton [14– +17], and gravity in interaction with electromagnetism and Abelian Higgs field [18]. In this current study, we investigate +the hairy extension of the Bertotti-Robinson (BR) spacetime [19–24] in the context of the Einstein-Maxwell theory [25], +a subgroup of hairy black holes which have gravity in interaction with electromagnetism and scalar fields [26]. This +new spacetime eventuates a black hole in closed space. One of the reasons to study a closed space in a cosmological +sense is that the energy and the topology of an open universe are difficult to be determined. Moreover, a static, closed +space can depict information related to the dynamics of the universe and an early universe [27]. +Furthermore, studying geodesics equations in the vicinity of black holes plays a great role in general relativity due to +obtaining important information on the structure of spacetime geometry. In this sense, we are going to study geodesics +equations for a test particle in a closed S3 black hole powered by pure magnetic fields introduced in [25]. The main +streamline of the present work is to investigate the trajectory of massless/null and massive/timelike test particles. We +derive the energy and angular momentum of the test particles and we illustrate some observable quantities in graphs +to discover the particle’s behavior in different circumstances. +II. +A REVIEW ON THE HAIRY EXTENSION OF THE BR SPACETIME +The black hole solution reported in [25] has been obtained in the context of Einstein-Maxwell-Scalar theory where +the action is given by (8πG = c = 1) +I = 1 +2 +� +d4x√−g +� +R − 2∂µψ∂µψ + V0 cos +� ψ +√ +2 +� +FµνF µν +� +(1) +in which ψ is the scalar field, V0 is a positive coupling constant, R is the Ricci scalar and FµνF µν is Maxwell’s +invariant. The black hole solution +ds2 = −f (u) dt2 + du2 +f (u) + R (u) 2dΩ2, +(2) +with the metric functions +∗Electronic address: vahideh.memari@emu.edu.tr +†Electronic address: habib.mazhari@emu.edu.tr +arXiv:2301.11665v1 [gr-qc] 27 Jan 2023 + +2 +f (u) = (u + uh) 3 (u − uh) +R2 +0u2 +, +(3) +and +R (u) = R0 +� +u +u + uh +� +, +(4) +the radial magnetic field +B (u) = P (u + uh)2 +R2 +0u2 +, +(5) +and the scalar field +ψ (u) = ±2 +√ +2 arctan +�� u +uh +� +, +(6) +fully satisfy the field equations. Herein, P is the magnetic monopole charge of the black hole, R2 +0 = P 2V0 and uh is +the event horizon. The scalar field (6) becomes constant as uh → 0 i.e., +ψ (u) → ± +√ +2π as uh → 0 +(7) +and consequently +V0 cos +� ψ +√ +2 +� +FµνF µν → −V0FµνF µν +(8) +which upon setting V0 = 1 the action reduces to Einstein-Maxwell’s, however, the spacetime becomes +ds2 = − u2 +P 2 dt2 + P 2du2 +u2 ++ P 2dΩ2. +(9) +The transformations u = 1 +r and t = P ˜t yield +ds2 = P 2 +r2 +� +−dt2 + dr2 + r2dΩ2� +(10) +which is nothing but the well-known BR spacetime [19, 20]. Although BR spacetime with topology R2 × S2 is not +a black hole and is regular everywhere, its hairy extension (2) is a singular black hole with two parameters i.e., R0 +and uh. In addition, this black hole is confined spatially to a S3−sphere of radius R0 which corresponds to u → ∞. +In other words, in the limit u → ∞ the spacetime behaves the same as the BR spacetime. Such a black hole i.e., +being closed in an S3 space is known in the literature as a closed black hole [28–33]. Some of the basic properties +of this closed black hole have been investigated in Ref. [25], however, the geodesics of this spacetime have not been +investigated. +III. +TRAJECTORIES OF PARTICLES +In this section, we study the geodesics of particles in the spacetime of the closed black hole introduced in Ref. [25]. +Precisely, we investigate the radial and circular time-like and null geodesics of test particles. The static spherically +symmetric closed black hole found in Ref. [25] is described by the line element (2) where the two metric functions +are given by (2) and (3) in which u ∈ [0, ∞) , R0 is a real constant and uh is the radius of the event horizon. The +Lagrangian of a massive particle with a unit mass is described by L = 1 +2gµν ˙xµ ˙xν which explicitly reads +L = −1 +2f (u) ˙t2 + +˙u2 +2f (u) + 1 +2R (u) 2 � +˙θ2 + sin2 θ ˙φ2� +(11) + +3 +in which a dot indicates a derivative with respect to an affine parameter (here λ) for massless particles and the proper +time for massive particles. The Lagrangian (11) is independent of time t and azimuthal angle φ which results in two +conserved quantities: 1) the energy E = − ∂L +∂ ˙t and 2) the angular momentum in φ direction ℓ = ∂L +∂ ˙φ . Therefore, one +writes +E = f (u) dt +dλ +(12) +and +ℓ = R (u)2 sin2 θdφ +dλ +(13) +which are both conserved. Since we are interested in the geodesics on the equatorial plane where θ = +π +2 , the angular +momentum reduces to +ℓ = R (u) 2 dφ +dλ. +(14) +Furthermore, considering the definition of the four-velocity U µ = dxµ +dλ that satisfies U µUµ = −ϵ in which ϵ = +1, −1, +and 0 for timelike, spacelike, and null particles, we explicitly obtain the condition +gµν +dxµ +dλ +dxν +dλ = −ϵ. +(15) +Upon combining Eqs. (12), (13) and (15), after setting θ = +π +2 , we determine +�du +dλ +�2 += E2 − f (u) +� +ϵ + +ℓ2 +R(u)2 +� +(16) +which is the geodesic equation for the radial coordinate u. In addition, by introducing an effective potential in the +form V 2 +eff (u) = +1 +2f (u) +� +ϵ + +ℓ2 +R(u)2 +� +and an effective energy ε2 +eff = +1 +2E2, the radial geodesics equation is given in a +more familiar form of an equation of motion for a test particle with unit mass, i.e., +1 +2 +�du +dλ +�2 ++ V 2 +eff (u) = E2 +eff. +(17) +The exact form of the effective potential by replacing f(u) from Eq. (3) is expressed by +V 2 +eff (u) = 1 +2 +(u + uh) 2 � +u2 − u2 +h +� +R2 +0u2 +� +ϵ + ℓ2 (u + uh)2 +R2 +0u2 +� +. +(18) +Let us note that, from Eq. (17) to have u a real coordinate, the condition E2 +eff ≥ V 2 +eff (u) should hold. Moreover, by +applying the chain rule du +dλ = +du +dφ +dφ +dλ, one eliminates the affine parameter from the main geodesic equation to get +�du +dφ +�2 += 2R (u) 4 +ℓ2 +� +E2 +eff − V 2 +eff (u) +� +. +(19) +The latter equation explicitly reads +�du +dφ +�2 += E2 +ℓ2 R (u) 4 − f (u) +� +ϵR(u)4 +ℓ2 ++ R(u)2 +� +:= G (u) +(20) +in which the right-hand side has to satisfy G (u) ≥ 0. In what follows we classify the geodesics in different cases. + +4 +A. +Radial Motion +For a radial motion the angular momentum has to be zero i.e., ℓ = 0, upon which (13) yields φ = const. such that +the particle will move radially. Hence, Eq. (16) reduces to +�du +dλ +�2 += E2 − ϵf (u) +(21) +In the following subsections, we shall investigate null and time-like geodesics for radial motion separately. +1. +Null geodesics ϵ = 0 +Considering ϵ = 0 in (21) for the null geodesics which describes the motion of a massless particle (photon), we +simply find +�du +dλ +�2 += E2. +(22) +On the other hand combining (12) and (22) we get, +du +dt = ±f (u) = ±(u + uh) 2 � +u2 − u2 +h +� +R2 +0u2 +. +(23) +Eq. (23) is integrable and explicitly yields the following +R2 +0 +4 +� 1 +2uh +ln +�(u − uh) (u0 + uh) +(u + uh) (u0 − uh) +� +− 3u + 2uh +(u + uh) 2 + 3u0 + 2uh +(u0 + uh) 2 +� += ± (t − t0) +(24) +where t0 is the initial time, t is the time measured by the distant observer and u0 is the initial position of the massless +particle (photon). Introducing u = uhx, u0 = uhx0 and T = 8uh +R2 +0 (t − t0) we obtain +T± = ± +� +ln +�(x − 1) (x0 + 1) +(x + 1) (x0 − 1) +� +− 2 3x + 2 +(x + 1) 2 + 2 3x0 + 2 +(x0 + 1) 2 +� +. +(25) +We note that ± reefers to the outgoing or ingoing light rays. In Fig. 1 we plot T+ in terms of x for various values +of x0. From this figure, we observe that on the equilateral plane, the photon moves away from the horizon toward +the boundary of the spacetime where u → ∞. The time needed for the photon to reach the boundary of the closed +spacetime is found to be +T+∞ = 2 3x0 + 2 +(x0 + 1) 2 + ln x0 + 1 +x0 − 1 +(26) +which is clearly finite. Furthermore, in Fig. 2 we plot T− in terms of x for various values of x0. We see that with initial +velocity toward the horizon, the photon reaches to the horizon at an infinite time measured by a distant observer. +2. +Time-like geodesics ϵ = 1 +Time-like geodesics refers to the motion of a massive particle where ϵ = 1 upon which (21) becomes +�du +dλ +�2 += E2 − (u + uh) 2 � +u2 − u2 +h +� +R2 +0u2 +. +(27) +Derivative of (27) with respect to the affine parameter λ implies + +5 +FIG. 1: The plots of T+ in terms of x, from Eq. (25) for x0 = 1.5, 2.0, ..., 5 with equal steps. +FIG. 2: The plots of T− in terms of x, from Eq. (25) for x0 = 1.5, 2.0, ..., 5 with equal steps. +d2u +dτ 2 = − +1 +R2 +0u3 (u + uh) +� +u3 + u3 +h +� +, +(28) +in which we set λ = τ with τ be the proper time. Obviously, the radial force per unit mass is attractive and toward +the horizon of the black hole. We assume that the particle is initially at rest located at u = u0 such that (27) yields + +4 +T ++ +3 +2 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +2010 +T +8 +6 +4 +2 +0 +1 +2 +3 +4 +5 +66 +FIG. 3: The generic plot of R2 +0 +uh V 2 +eff (u) in terms of x = +u +uh . The shaded region implies inside the black hole and the horizon is at x = 1. +The massive particle in its radial motion has no chance to escape to the edge of spacetime. This means that the massive timelike +particle either directly collapses to the singularity of the spacetime or after it bounces from the potential barrier, as shown in the figure. +E2 = (u0 + uh) 2 � +u2 +0 − u2 +h +� +R2 +0u2 +0 +(29) +upon which (27) becomes +�du +dτ +�2 += 1 +R2 +0 +� +(u0 + uh) 2 � +u2 +0 − u2 +h +� +u2 +0 +− (u + uh) 2 � +u2 − u2 +h +� +u2 +� +, +(30) +and with ℓ = 0 and ϵ = 1, the effective potential becomes +V 2 +eff (u) = 1 +2 +(u + uh) 2 � +u2 − u2 +h +� +R2 +0u2 +. +(31) +In Fig. 3 we plotted R2 +0 +uh V 2 +eff (u) in terms of x = +u +uh which shows that the potential is an increasing function implying +an attractive force toward the singularity. Therefore no matter what is the energy of the particle, its fate is a collapse +into the singularity. Finally, using Eqs. (27) and (12) we find the radial equation of motion of a massive particle in +terms of the observer time i.e., +�du +dt +�2 += f (u) 2 − f (u) 3 +E2 +(32) +and explicitly +�du +dt +�2 += (u + uh) 4 � +u2 − u2 +h +� 2 +R4 +0u4 +� +1 − (u + uh) 2 � +u2 − u2 +h +� +E2R2 +0u2 +� +. +(33) + +10 +5 +29 +2 +R +0 +2 +5 +-107 +FIG. 4: The radial motion of a massive particle (dashed-blue curve) with ˜R0 = 1, +� +dx +dt +� +t=0 = 0 and ˜E2 = 375 +16 . The horizon is located at +x = 1 (red solid line). This is the numeric plot of the radial geodesic equation (36) constraint by (35). +Considering the timelike particle is initially at rest where u = u0, one writes +0 = (u0 + uh) 4 � +u2 +0 − u2 +h +� 2 +R4 +0u4 +0 +� +1 − (u0 + uh) 2 � +u2 +0 − u2 +h +� +E2R2 +0u2 +0 +� +(34) +which yields the conserved energy of the particle in terms of its initial position given in Eq. +(29). +Introducing +u (t) = x (t) uh, R0 = ˜R0√uh and E = ˜E√uh we obtain the following equation +�dx +dt +�2 += +� +x2 − 1 +�2 (1 + x)4 +˜R4 +0x4 +− +� +x2 − 1 +�3 (1 + x)6 +˜R6 +0 ˜E2x6 +(35) +and after differentiating with respect to t the main differential equation is obtained +d2x +dt2 = 2 +� +x2 − 1 +� +(1 + x)4 � +1 − x + x2� +˜R4 +0x5 +− 3 +� +x2 − 1 +�2 (1 + x)6 � +1 − x + x2� +˜R6 +0 ˜E2x7 +. +(36) +The latter equation is highly nonlinear and cannot be solved analytically. Therefore we solve it by applying the +numerical method. In Fig. 4 we plotted x (t) in terms of t where the parameters are set to be ˜R0 = 1 and ˜E2 = 375 +16 +and the initial conditions are given by x0 = 4 and +� dx +dt +� +0 = 0. +B. +Circular motion +In this section, we investigate the circular motion of a massless/massive particle. First of all we impose du +dφ +��� +u=uc = 0 +in which uc is the radius of the circular orbit. Therefore, (20) implies +E2 +ℓ2 R (uc) 4 − f (uc) +� +ϵR (uc) 4 +ℓ2 ++ R (uc) 2 +� += 0 +(37) + +4 +3 +2 +0 +七 +0 +0.5 +1.5 +1 +28 +FIG. 5: Trajectory of a photon (dashed-blue curve) with ˜ℓ2 = 2, +� +dx +dϕ +� +ϕ=0 = 0 and ˜E2 = 9375 +128 . The horizon is located at x = 1 (red +circle). +Furthermore, for an equilibrium orbit one should impose d2u +dφ2 = 0. From (20) one obtains +d2u +dφ2 = 1 +2 +d +du +�E2 +ℓ2 R (u) 4 − f (u) +� +ϵR (u) 4 +ℓ2 ++ R (u) 2 +�� +(38) +upon which d2u +dφ2 = 0 yields +d +du +�E2 +ℓ2 R (u) 4 − f (u) +� +ϵR (u) 4 +ℓ2 ++ R (u) 2 +������ +u=uc += 0. +(39) +Applying the conditions (37) and (39) one gets +ℓ2 = − +R2 +0ϵu2 +c +� +u2 +c − ucuh + u2 +h +� +(u2c − 2ucuh + 2u2 +h) (uc + uh) 2 +(40) +and +E2 = −ϵuh (uc − uh) 2 (uc + uh) 3 +R2 +0u2c (u2c − 2ucuh + 2u2 +h) +(41) +Obviously, as the left hand sides of Eq. (40) and Eq. (41) are positive which is in contrast with the right hand side +we understand that there is no stable circular motion for this closed black hole. +C. +General equatorial motion +Having known that a circular motion is not possible, in this section we study the general motion of null/timelike +particles on the equatorial plane. The trajectory of such particles has to satisfy the following equations +� dx +dϕ +�2 += ϵ(1 − x) x2 +(1 + x) ˜ℓ2 + +˜E2x4 +˜ℓ2 (1 + x)4 + +� +1 − x2� +(42) + +-2 +0 +2 +3 +4 +59 +FIG. 6: Trajectory of a timelike particle (dashed-blue curve) with ˜ℓ2 = 2, +� +dx +dϕ +� +ϕ=0 = 0 and ˜E2 = 12375 +128 . The horizon is located at x = 1 +(red circle). +and +d2x +dϕ2 = ϵ +� +1 − x − x2� +x +(1 + x)2 ˜ℓ2 ++ +2 ˜E2x3 +˜ℓ2 (1 + x)5 − x +(43) +in which we have introduced x (ϕ) = u (ϕ) /uh, ˜ℓ2 = ℓ2/R2 +0, and ˜E2 = R2 +0E2/u2 +h. The nonlinear structure of the +geodesics main equation (43) prevents obtaining an exact solution, however, we solved (43) using a numerical method, +and the results are presented in Figs. 5 and 6 for the null (ϵ = 0) and the timelike (ϵ = 1) particles. This is observed +that with nonzero angular momentum, no matter what the initial conditions are, the particle eventually falls into the +black hole. +IV. +CONCLUSION +In this paper, we studied the geodesics of the null and timelike particles in the spacetime of the closed black hole +introduced in [25]. This black hole is a hairy extension of the regular BR spacetime which has been found very +recently in the theory of gravity coupled with a scalar field and the Maxwell electrodynamics described by the action +(1). In the first part, we studied the radial motion of a photon and a massive particle on the equatorial plane. For a +photon, we observed that depending on light beam’s direction it may go to the edge of the universe in a finite time +measured by a distant observer or it reaches the black hole in an infinite time. On the other hand, a massive timelike +particle collapses to the singularity irrespective of the direction of its initial velocity. Furthermore, we have explicitly +shown that there is no stable circular orbit neither for a photon nor for a massive particle. This study sheds light on +the physical properties of the black hole of Ref. [25] which is of a class of black holes called closed black holes. We +observed that massive particles as well as non-radially outgoing light beams all are attracted by the black hole. Is +this unusual behavior a universal behavior of all closed black holes? This question can not be answered here. One +has to study the geodesics of other closed black holes and a concrete answer may need further investigation. We leave +this problem open. +[1] W. Israel, Phys. Rev. 164, 1776 (1967). + +-2 +0 +2 +3 +4 +510 +[2] W. Israel, Commun. Math. Phys. 8, 245 (1968). +[3] B. Carter, Phys. Rev. Lett. 26, 1653 (1971). +[4] R. M. Wald, Phys. Rev. Lett. 26, 1653 (1971). +[5] J. D. Bekenstein, Phys. Rev. Lett. D 51, R6608 (1995). +[6] M. S. Volkov and D. V. Galtsov, JETP Lett. 50, 346 (1989). +[7] M. S. Volkov and D. V. Galtsov, Pisma Zh. Eksp. Teor. Fiz. 50, 312 (1989). +[8] P. Bizon, Phys. Rev. Lett. 64, 2844 (1990). +[9] B. R. Greene, S. D. Mathur and C. M. O’Neill, Phys. Rev. D 47, 2242 (1993). +[10] P. Nicolini and E. Spallucci, Class. Quantum Grav. 27, 015010 (2010). +[11] T. J. Allen, M. J. Bowick and A. Lahiri, Phys. Lett. B 237, 47 (1990). +[12] B. A. Campbell, N. Kaloper and K. A. Olive, Phys. Lett. Rev. B 263, 346 (1991). +[13] K. M. Lee and E. J. Weinberg, Phys. Rev. D 44, 3159 (1991). +[14] G. W. Gibbons and K.I. Maeda, Nucl. Phys. B 298, 741 (1988). +[15] I. Ichinose and H. Yamazaki, Mod. Phys. Lett. A 4, 1509 (1989). +[16] H. Yamazaki and I. Ichinose, Class. Quntum Grav. 9, 257 (1992). +[17] D. Grafinkle, G. T. Horowitz and A. Strominger, Phys. Rev. D 43, 3140 (1992), [ Erratum-ibid D 45, 3888] +[18] F. Dowker, R. Grogory and J. H. Traschen, Phys. Rev D 45, 2762 (1992). +[19] B. Bertotti, Phys. Rev. 116, 1331 (1959). +[20] I. Robinson, Bull. Acad. Sci. Polon. 7, 351 (1959). +[21] H. Stephani, Commun. Math. Phys. 5, 337 (1967). +[22] P. Dolan, Commun. Math. Phys. 9, 161 (1968). +[23] N. Tariq and B. O. J. Tupper, J. Math. Phys. 15, 2232 (1974). +[24] D. Garfinkle, E. N. Glass, Class. Quantum Grav. 28, 215012 (2011). +[25] S. H. Mazharimousavi, Class. Quantum Grav. 39, 167001 (2022). +[26] P. A. Gonzalez, E. Papantonopoulos, J. Saavedra, Y. Vasquez, J. High Energ. Phys. 11, 1 (2014). +[27] I. Cho and H. C. Kim, Phys. Rev. D 95, 084052 (2017). +[28] K. A. Bronnikov and O. B. Zaslavskii, Phys. Rev. D 78, 021501(R) (2008). +[29] K. A. Bronnikov and O. B. Zaslavskii, Class. Quantum Grav. 26, 165004 (2009). +[30] K. A. Bronnikov, E. Elizalde, S. D. Odintsov and O. B. Zaslavskii, Phys. Rev. D 78, 064049 (2008). +[31] I. Cho, Eur. Phys. J. C 79, 42 (2019). +[32] I. Cho and H.-C. Kim, Phys. Rev. D 95, 084052 (2017). +[33] H.-C. Kim, Phys. Rev. D 96, 064053 (2017). + diff --git a/WNFJT4oBgHgl3EQf4S11/content/tmp_files/load_file.txt b/WNFJT4oBgHgl3EQf4S11/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6e3ea55c575eef02187ecac6c60bf7d7d897417d --- /dev/null +++ b/WNFJT4oBgHgl3EQf4S11/content/tmp_files/load_file.txt @@ -0,0 +1,376 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf,len=375 +page_content='Hairy extension of the Bertotti-Robinson swallows almost everything Vahideh Memari∗ and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Habib Mazharimousavi† Department of Physics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 10, Turkey (Dated: January 30, 2023) A hairy extension of the Bertotti-Robinson regular spacetime has been recently introduced in the context of the Einstein-Maxwell-Scaler theory that surprisingly is a closed singular black hole [CQG39(2022)167001].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' We investigate the geodesics of the null and timelike particles in this space- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' We show that in the radial motion on the equatorial plane, while photons may collapse to the singularity or escape to the edge of the universe a massive particle always collapses to the singular- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Also, it is proven that on the equatorial plane there is no stable orbit not only for photons but also for massive particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' However, the general geodesics of null and massive particles reveal that all particles except the outgoing light ray, eventually fall into the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' This unique feature makes closed black holes interesting for further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' INTRODUCTION In general relativity, no-hair theorems state that black holes are described by only three parameters, their mass M, their electromagnetic charge Q, and their angular momentum ℓ [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Recent studies on black holes indicate the existence of hairy black holes some of which form in the presence of Yang-Mills fields [6–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Hairy black holes are solutions to Einstein’s equations in the interaction of diverse kinds of matter fields with gravity [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' For instance gravity in interaction with electromagnetism and axion [11–13], gravity coupled to electromagnetism and dilaton [14– 17], and gravity in interaction with electromagnetism and Abelian Higgs field [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' In this current study, we investigate the hairy extension of the Bertotti-Robinson (BR) spacetime [19–24] in the context of the Einstein-Maxwell theory [25], a subgroup of hairy black holes which have gravity in interaction with electromagnetism and scalar fields [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' This new spacetime eventuates a black hole in closed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' One of the reasons to study a closed space in a cosmological sense is that the energy and the topology of an open universe are difficult to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Moreover, a static, closed space can depict information related to the dynamics of the universe and an early universe [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Furthermore, studying geodesics equations in the vicinity of black holes plays a great role in general relativity due to obtaining important information on the structure of spacetime geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' In this sense, we are going to study geodesics equations for a test particle in a closed S3 black hole powered by pure magnetic fields introduced in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' The main streamline of the present work is to investigate the trajectory of massless/null and massive/timelike test particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' We derive the energy and angular momentum of the test particles and we illustrate some observable quantities in graphs to discover the particle’s behavior in different circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' A REVIEW ON THE HAIRY EXTENSION OF THE BR SPACETIME The black hole solution reported in [25] has been obtained in the context of Einstein-Maxwell-Scalar theory where the action is given by (8πG = c = 1) I = 1 2 � d4x√−g � R − 2∂µψ∂µψ + V0 cos � ψ √ 2 � FµνF µν � (1) in which ψ is the scalar field, V0 is a positive coupling constant, R is the Ricci scalar and FµνF µν is Maxwell’s invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' The black hole solution ds2 = −f (u) dt2 + du2 f (u) + R (u) 2dΩ2, (2) with the metric functions ∗Electronic address: vahideh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='memari@emu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='tr †Electronic address: habib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='mazhari@emu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='tr arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='11665v1 [gr-qc] 27 Jan 2023 2 f (u) = (u + uh) 3 (u − uh) R2 0u2 , (3) and R (u) = R0 � u u + uh � , (4) the radial magnetic field B (u) = P (u + uh)2 R2 0u2 , (5) and the scalar field ψ (u) = ±2 √ 2 arctan �� u uh � , (6) fully satisfy the field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Herein, P is the magnetic monopole charge of the black hole, R2 0 = P 2V0 and uh is the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' The scalar field (6) becomes constant as uh → 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=', ψ (u) → ± √ 2π as uh → 0 (7) and consequently V0 cos � ψ √ 2 � FµνF µν → −V0FµνF µν (8) which upon setting V0 = 1 the action reduces to Einstein-Maxwell’s, however, the spacetime becomes ds2 = − u2 P 2 dt2 + P 2du2 u2 + P 2dΩ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (9) The transformations u = 1 r and t = P ˜t yield ds2 = P 2 r2 � −dt2 + dr2 + r2dΩ2� (10) which is nothing but the well-known BR spacetime [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Although BR spacetime with topology R2 × S2 is not a black hole and is regular everywhere, its hairy extension (2) is a singular black hole with two parameters i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=', R0 and uh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' In addition, this black hole is confined spatially to a S3−sphere of radius R0 which corresponds to u → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' In other words, in the limit u → ∞ the spacetime behaves the same as the BR spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Such a black hole i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=', being closed in an S3 space is known in the literature as a closed black hole [28–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Some of the basic properties of this closed black hole have been investigated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' [25], however, the geodesics of this spacetime have not been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' TRAJECTORIES OF PARTICLES In this section, we study the geodesics of particles in the spacetime of the closed black hole introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Precisely, we investigate the radial and circular time-like and null geodesics of test particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' The static spherically symmetric closed black hole found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' [25] is described by the line element (2) where the two metric functions are given by (2) and (3) in which u ∈ [0, ∞) , R0 is a real constant and uh is the radius of the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' The Lagrangian of a massive particle with a unit mass is described by L = 1 2gµν ˙xµ ˙xν which explicitly reads L = −1 2f (u) ˙t2 + ˙u2 2f (u) + 1 2R (u) 2 � ˙θ2 + sin2 θ ˙φ2� (11) 3 in which a dot indicates a derivative with respect to an affine parameter (here λ) for massless particles and the proper time for massive particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' The Lagrangian (11) is independent of time t and azimuthal angle φ which results in two conserved quantities: 1) the energy E = − ∂L ∂ ˙t and 2) the angular momentum in φ direction ℓ = ∂L ∂ ˙φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Therefore, one writes E = f (u) dt dλ (12) and ℓ = R (u)2 sin2 θdφ dλ (13) which are both conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Since we are interested in the geodesics on the equatorial plane where θ = π 2 , the angular momentum reduces to ℓ = R (u) 2 dφ dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (14) Furthermore, considering the definition of the four-velocity U µ = dxµ dλ that satisfies U µUµ = −ϵ in which ϵ = +1, −1, and 0 for timelike, spacelike, and null particles, we explicitly obtain the condition gµν dxµ dλ dxν dλ = −ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (15) Upon combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (12), (13) and (15), after setting θ = π 2 , we determine �du dλ �2 = E2 − f (u) � ϵ + ℓ2 R(u)2 � (16) which is the geodesic equation for the radial coordinate u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' In addition, by introducing an effective potential in the form V 2 eff (u) = 1 2f (u) � ϵ + ℓ2 R(u)2 � and an effective energy ε2 eff = 1 2E2, the radial geodesics equation is given in a more familiar form of an equation of motion for a test particle with unit mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=', 1 2 �du dλ �2 + V 2 eff (u) = E2 eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (17) The exact form of the effective potential by replacing f(u) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (3) is expressed by V 2 eff (u) = 1 2 (u + uh) 2 � u2 − u2 h � R2 0u2 � ϵ + ℓ2 (u + uh)2 R2 0u2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (18) Let us note that, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (17) to have u a real coordinate, the condition E2 eff ≥ V 2 eff (u) should hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Moreover, by applying the chain rule du dλ = du dφ dφ dλ, one eliminates the affine parameter from the main geodesic equation to get �du dφ �2 = 2R (u) 4 ℓ2 � E2 eff − V 2 eff (u) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (19) The latter equation explicitly reads �du dφ �2 = E2 ℓ2 R (u) 4 − f (u) � ϵR(u)4 ℓ2 + R(u)2 � := G (u) (20) in which the right-hand side has to satisfy G (u) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' In what follows we classify the geodesics in different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Radial Motion For a radial motion the angular momentum has to be zero i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=', ℓ = 0, upon which (13) yields φ = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' such that the particle will move radially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Hence, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (16) reduces to �du dλ �2 = E2 − ϵf (u) (21) In the following subsections, we shall investigate null and time-like geodesics for radial motion separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Null geodesics ϵ = 0 Considering ϵ = 0 in (21) for the null geodesics which describes the motion of a massless particle (photon), we simply find �du dλ �2 = E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (22) On the other hand combining (12) and (22) we get, du dt = ±f (u) = ±(u + uh) 2 � u2 − u2 h � R2 0u2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (23) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (23) is integrable and explicitly yields the following R2 0 4 � 1 2uh ln �(u − uh) (u0 + uh) (u + uh) (u0 − uh) � − 3u + 2uh (u + uh) 2 + 3u0 + 2uh (u0 + uh) 2 � = ± (t − t0) (24) where t0 is the initial time, t is the time measured by the distant observer and u0 is the initial position of the massless particle (photon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Introducing u = uhx, u0 = uhx0 and T = 8uh R2 0 (t − t0) we obtain T± = ± � ln �(x − 1) (x0 + 1) (x + 1) (x0 − 1) � − 2 3x + 2 (x + 1) 2 + 2 3x0 + 2 (x0 + 1) 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (25) We note that ± reefers to the outgoing or ingoing light rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 1 we plot T+ in terms of x for various values of x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' From this figure, we observe that on the equilateral plane, the photon moves away from the horizon toward the boundary of the spacetime where u → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' The time needed for the photon to reach the boundary of the closed spacetime is found to be T+∞ = 2 3x0 + 2 (x0 + 1) 2 + ln x0 + 1 x0 − 1 (26) which is clearly finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Furthermore, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 2 we plot T− in terms of x for various values of x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' We see that with initial velocity toward the horizon, the photon reaches to the horizon at an infinite time measured by a distant observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Time-like geodesics ϵ = 1 Time-like geodesics refers to the motion of a massive particle where ϵ = 1 upon which (21) becomes �du dλ �2 = E2 − (u + uh) 2 � u2 − u2 h � R2 0u2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (27) Derivative of (27) with respect to the affine parameter λ implies 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 1: The plots of T+ in terms of x, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (25) for x0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=', 5 with equal steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 2: The plots of T− in terms of x, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (25) for x0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=', 5 with equal steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' d2u dτ 2 = − 1 R2 0u3 (u + uh) � u3 + u3 h � , (28) in which we set λ = τ with τ be the proper time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Obviously, the radial force per unit mass is attractive and toward the horizon of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' We assume that the particle is initially at rest located at u = u0 such that (27) yields 4 T + 3 2 0 2 4 6 8 10 12 14 16 18 2010 T 8 6 4 2 0 1 2 3 4 5 66 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 3: The generic plot of R2 0 uh V 2 eff (u) in terms of x = u uh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' The shaded region implies inside the black hole and the horizon is at x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' The massive particle in its radial motion has no chance to escape to the edge of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' This means that the massive timelike particle either directly collapses to the singularity of the spacetime or after it bounces from the potential barrier, as shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' E2 = (u0 + uh) 2 � u2 0 − u2 h � R2 0u2 0 (29) upon which (27) becomes �du dτ �2 = 1 R2 0 � (u0 + uh) 2 � u2 0 − u2 h � u2 0 − (u + uh) 2 � u2 − u2 h � u2 � , (30) and with ℓ = 0 and ϵ = 1, the effective potential becomes V 2 eff (u) = 1 2 (u + uh) 2 � u2 − u2 h � R2 0u2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (31) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 3 we plotted R2 0 uh V 2 eff (u) in terms of x = u uh which shows that the potential is an increasing function implying an attractive force toward the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Therefore no matter what is the energy of the particle, its fate is a collapse into the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Finally, using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (27) and (12) we find the radial equation of motion of a massive particle in terms of the observer time i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=', �du dt �2 = f (u) 2 − f (u) 3 E2 (32) and explicitly �du dt �2 = (u + uh) 4 � u2 − u2 h � 2 R4 0u4 � 1 − (u + uh) 2 � u2 − u2 h � E2R2 0u2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (33) 10 5 29 2 R 0 2 5 107 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 4: The radial motion of a massive particle (dashed-blue curve) with ˜R0 = 1, � dx dt � t=0 = 0 and ˜E2 = 375 16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' The horizon is located at x = 1 (red solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' This is the numeric plot of the radial geodesic equation (36) constraint by (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Considering the timelike particle is initially at rest where u = u0, one writes 0 = (u0 + uh) 4 � u2 0 − u2 h � 2 R4 0u4 0 � 1 − (u0 + uh) 2 � u2 0 − u2 h � E2R2 0u2 0 � (34) which yields the conserved energy of the particle in terms of its initial position given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Introducing u (t) = x (t) uh, R0 = ˜R0√uh and E = ˜E√uh we obtain the following equation �dx dt �2 = � x2 − 1 �2 (1 + x)4 ˜R4 0x4 − � x2 − 1 �3 (1 + x)6 ˜R6 0 ˜E2x6 (35) and after differentiating with respect to t the main differential equation is obtained d2x dt2 = 2 � x2 − 1 � (1 + x)4 � 1 − x + x2� ˜R4 0x5 − 3 � x2 − 1 �2 (1 + x)6 � 1 − x + x2� ˜R6 0 ˜E2x7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (36) The latter equation is highly nonlinear and cannot be solved analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Therefore we solve it by applying the numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 4 we plotted x (t) in terms of t where the parameters are set to be ˜R0 = 1 and ˜E2 = 375 16 and the initial conditions are given by x0 = 4 and � dx dt � 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Circular motion In this section, we investigate the circular motion of a massless/massive particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' First of all we impose du dφ ��� u=uc = 0 in which uc is the radius of the circular orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Therefore, (20) implies E2 ℓ2 R (uc) 4 − f (uc) � ϵR (uc) 4 ℓ2 + R (uc) 2 � = 0 (37) 4 3 2 0 七 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content='5 1 28 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 5: Trajectory of a photon (dashed-blue curve) with ˜ℓ2 = 2, � dx dϕ � ϕ=0 = 0 and ˜E2 = 9375 128 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' The horizon is located at x = 1 (red circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Furthermore, for an equilibrium orbit one should impose d2u dφ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' From (20) one obtains d2u dφ2 = 1 2 d du �E2 ℓ2 R (u) 4 − f (u) � ϵR (u) 4 ℓ2 + R (u) 2 �� (38) upon which d2u dφ2 = 0 yields d du �E2 ℓ2 R (u) 4 − f (u) � ϵR (u) 4 ℓ2 + R (u) 2 ������ u=uc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (39) Applying the conditions (37) and (39) one gets ℓ2 = − R2 0ϵu2 c � u2 c − ucuh + u2 h � (u2c − 2ucuh + 2u2 h) (uc + uh) 2 (40) and E2 = −ϵuh (uc − uh) 2 (uc + uh) 3 R2 0u2c (u2c − 2ucuh + 2u2 h) (41) Obviously, as the left hand sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (40) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' (41) are positive which is in contrast with the right hand side we understand that there is no stable circular motion for this closed black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' General equatorial motion Having known that a circular motion is not possible, in this section we study the general motion of null/timelike particles on the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' The trajectory of such particles has to satisfy the following equations � dx dϕ �2 = ϵ(1 − x) x2 (1 + x) ˜ℓ2 + ˜E2x4 ˜ℓ2 (1 + x)4 + � 1 − x2� (42) 2 0 2 3 4 59 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 6: Trajectory of a timelike particle (dashed-blue curve) with ˜ℓ2 = 2, � dx dϕ � ϕ=0 = 0 and ˜E2 = 12375 128 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' The horizon is located at x = 1 (red circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' and d2x dϕ2 = ϵ � 1 − x − x2� x (1 + x)2 ˜ℓ2 + 2 ˜E2x3 ˜ℓ2 (1 + x)5 − x (43) in which we have introduced x (ϕ) = u (ϕ) /uh, ˜ℓ2 = ℓ2/R2 0, and ˜E2 = R2 0E2/u2 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' The nonlinear structure of the geodesics main equation (43) prevents obtaining an exact solution, however, we solved (43) using a numerical method, and the results are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 5 and 6 for the null (ϵ = 0) and the timelike (ϵ = 1) particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' This is observed that with nonzero angular momentum, no matter what the initial conditions are, the particle eventually falls into the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' CONCLUSION In this paper, we studied the geodesics of the null and timelike particles in the spacetime of the closed black hole introduced in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' This black hole is a hairy extension of the regular BR spacetime which has been found very recently in the theory of gravity coupled with a scalar field and the Maxwell electrodynamics described by the action (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' In the first part, we studied the radial motion of a photon and a massive particle on the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' For a photon, we observed that depending on light beam’s direction it may go to the edge of the universe in a finite time measured by a distant observer or it reaches the black hole in an infinite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' On the other hand, a massive timelike particle collapses to the singularity irrespective of the direction of its initial velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Furthermore, we have explicitly shown that there is no stable circular orbit neither for a photon nor for a massive particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' This study sheds light on the physical properties of the black hole of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' [25] which is of a class of black holes called closed black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' We observed that massive particles as well as non-radially outgoing light beams all are attracted by the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Is this unusual behavior a universal behavior of all closed black holes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' This question can not be answered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' One has to study the geodesics of other closed black holes and a concrete answer may need further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' We leave this problem open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Israel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 164, 1776 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 2 0 2 3 4 510 [2] W.' metadata={'source': 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R6608 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Volkov and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' Galtsov, JETP Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} +page_content=' 50, 346 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFJT4oBgHgl3EQf4S11/content/2301.11665v1.pdf'} 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--git a/XNFQT4oBgHgl3EQfczY1/content/tmp_files/2301.13328v1.pdf.txt b/XNFQT4oBgHgl3EQfczY1/content/tmp_files/2301.13328v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..02d779f1adb59634870708b3d46c6cf21b48d588 --- /dev/null +++ b/XNFQT4oBgHgl3EQfczY1/content/tmp_files/2301.13328v1.pdf.txt @@ -0,0 +1,1597 @@ +arXiv:2301.13328v1 [cs.AI] 30 Jan 2023 +On the Complexity of Enumerating Prime Implicants +from Decision-DNNF Circuits +Alexis de Colnet1 and Pierre Marquis1,2 +1Univ. Artois, CNRS, Centre de Recherche en Informatique de Lens (CRIL), F-62300 Lens, France +2Institut Universitaire de France +{decolnet, marquis}@cril.fr +Abstract +We consider the problem Enum·IP of enumerating +prime implicants of Boolean functions represented +by decision decomposable negation normal form +(dec-DNNF) circuits. We study Enum·IP from dec- +DNNF within the framework of enumeration com- +plexity and prove that it is in OutputP, the class +of output polynomial enumeration problems, and +more precisely in IncP, the class of polynomial in- +cremental time enumeration problems. We then fo- +cus on two closely related, but seemingly harder, +enumeration problems where further restrictions +are put on the prime implicants to be generated. In +the first problem, one is only interested in prime +implicants representing subset-minimal abductive +explanations, a notion much investigated in AI for +more than three decades. In the second problem, +the target is prime implicants representing suffi- +cient reasons, a recent yet important notion in the +emerging field of eXplainable AI, since they aim +to explain predictions achieved by machine learn- +ing classifiers. We provide evidence showing that +enumerating specific prime implicants correspond- +ing to subset-minimal abductive explanations or to +sufficient reasons is not in OutputP. +1 +Introduction +Prime implicants are a key concept when dealing with +Boolean functions since the notion has been introduced seven +decades ago [Quine, 1952]. +Within AI, prime implicants +(or the dual concept of prime implicates) have been con- +sidered for modeling and solving a number of problems, +including compiling knowledge [Reiter and de Kleer, 1987] +and generating explanations of various kinds. +This is +the case in logic-based abductive reasoning (see e.g., +[Selman and Levesque, 1990; +Eiter and Gottlob, 1995]), +a +form of inference required in a number of applications when +the available knowledge base is incomplete (e.g., in medicine) +and because of such an incompleteness, it cannot alone ex- +plain the observations that are made about the state of the +world. +Abduction gave rise to much research in AI for +the past three decades, especially because it is closely con- +nected to other reasoning settings, including truth main- +tenance [de Kleer, 1986], assumption-based reasoning and +closed-world reasoning (see e.g., [Marquis, 2000] for a sur- +vey). +Formally, the explanations one looks for are terms +over a preset alphabet (composed of the so-called abducible +variables, e.g., representing diseases) such that the manifes- +tations that are reported (e.g., some symptoms) are logical +consequences of the background knowledge when completed +by such a term. In order to avoid trivial explanations, one +also asks those terms to be consistent with the knowledge +base. Explanations that are the less demanding ones from +a logical standpoint (i.e., subset-minimal ones) can be char- +acterized as specific prime implicants. More recently, de- +riving explanations justifying why certain predictions have +been made has appeared as essential for ensuring trust- +worthy Machine Learning (ML) technologies [Miller, 2019; +Molnar, 2019]. In the research area of eXplainable AI (XAI), +recent work has shown how ML classifiers of various types +(including black boxes) can be associated with Boolean +circuits (alias transparent or “white” boxes), exhibiting +the same input-output behaviours [Narodytska et al., 2018; +Shih et al., 2018a; Shih et al., 2019]. Thanks to such map- +pings, XAI queries about classifiers can be delegated to the +corresponding circuits. +The notion of sufficient reason of +an instance given a Boolean function f modeling a binary +classifier has been introduced in [Darwiche and Hirth, 2020]. +Given an instance a (a simply is an assignment, i.e., a vec- +tor of truth values given to each of the n variables) such that +f(a) = 1 (resp. f(a) = 0), a sufficient reason for a is a +subset-minimal partial assignment a′ which is coherent with +a (i.e., a and a′ give the same values to the variables that are +assigned in a′) and which satisfies the property that for every +extension a′′ of a′ we have f(a′′) = 1 (resp. f(a′′) = 0). +The features assigned in a′ (and the way they are assigned) +can be viewed as explaining why a has been classified by f +as a positive (or as a negative) instance. +Whatever the way prime implicants are used, generating +them is in general a computationally demanding task, for at +least two reasons. On the one hand, deriving a single prime +implicant of a Boolean function represented by a proposi- +tional formula (or circuit) is NP-hard since such a formula +is satisfiable when it has a prime implicant, and it is valid +precisely when this prime implicant is the empty term. On +the other hand, a source of complexity is the number of prime +implicants that may prevent from computing them all. In- + +deed, it is well-known that the number of prime implicants +of a Boolean function can be exponential in the number of +variables of the function, and, for many representations of +the function, also exponential in the size of the representa- +tion (just consider the parity function as a matter of example). +In more detail, the number of prime implicants of a Boolean +function can be larger than the number of assignments sat- +isfying the function [Dunham and Fridshal, 1959]; there also +exist families of Boolean functions over n variables having +Ω( 3n +n ) prime implicants [Chandra and Markowsky, 1978]. +In this paper, we focus on the issue of enumerating prime +implicants of a Boolean function represented by a decision +decomposable negation normal form circuit (alias a dec- +DNNF circuit). The question is to determine whether such +prime implicants can be enumerated “efficiently”, which is +obviously not the case when the circuit considered is uncon- +strained (as explained above, in such a case, computing a sin- +gle prime implicant is already hard). This question is impor- +tant for all the problems listed previously, when prime impli- +cants represent explanations: since they are typically too nu- +merous to be computed as a whole, it makes sense to derive +them in an incremental way, with some performance guar- +antees in the generation; this lets the user who asked for an +explanation deciding what to do after each derivation, namely +to stop the enumeration process since he/she is satisfied by the +explanation that has been provided, or alternatively to ask for +a further explanation. +The dec-DNNF language [Oztok and Darwiche, 2014; +Darwiche, 2001] and its subsets FBDD (free binary deci- +sion diagrams) [Gergov and Meinel, 1994], OBDD (ordered +binary decision diagrams) [Bryant, 1986] and even DT (the +set of all binary decision trees over Boolean variables, see +e.g., +[Wegener, 2000, Chapter 2]) appear at first sight as +good candidates for representing the function in the perspec- +tive of enumerating “efficiently” its prime implicants. In- +deed, they are known as tractable representation languages +(they support in polynomial time many queries and trans- +formations from the so-called knowledge compilation map +[Darwiche and Marquis, 2002; Koriche et al., 2013]). +The main contribution of the paper is as follows. +We +give a polynomial incremental time algorithm for enumer- +ating the prime implicants of a Boolean function f repre- +sented by a dec-DNNF circuit Σ. Given Σ and a positive +integer k, this algorithm returns k prime implicants of Σ in +O(poly(k + |Σ|)) time, or returns all prime implicants of Σ +if there are fewer than k. This shows that enumerating prime +implicants from dec-DNNF is in the enumeration complex- +ity class IncP [Strozecki, 2019]. We also provide evidence +showing that enumerating specific prime implicants corre- +sponding to subset-minimal abductive explanations or to suf- +ficient reasons is not in OutputP: on the one hand, comput- +ing a single subset-minimal abductive explanation from an +OBDD circuit or a decision tree is NP-hard; on the other +hand, the existence of an output polynomial time algorithm +for enumerating sufficient reasons from an OBDD circuit or +a decision tree would lead to an output polynomial time al- +gorithm for enumerating the minimal transversals of a hy- +pergraph, thus answering a long-standing question related to +monotone dualization [Eiter et al., 2008]. +The rest of the paper is organized as follows. We start with +some preliminaries (Section 2) where the language of dec- +DNNF circuits and the framework used to study enumera- +tion problems are presented. We formally define the problem +Enum·IP of enumerating prime implicants. Then in Section 3 +we show that generating the set of all prime implicants from +a dec-DNNF circuit is feasible in output polynomial time. +From there, we show in Section 4 that Enum·IP from dec- +DNNF is in fact in IncP and point out a polynomial incremen- +tal time enumeration algorithm. Finally, in Section 5 we focus +on subset-minimal abductive explanations and sufficient rea- +sons and show that for each of the two cases, the enumeration +issue is seemingly harder than in the case when all prime im- +plicants are considered. All the proofs are reported in a final +appendix. +2 +Preliminaries +A Boolean function over n variables x1, . . . , xn is a mapping +f from {0, 1}n to {0, 1}. The set of variables of f is denoted +by var(f). The assignments to var(f) mapped to 1 by f are +called satisfying assignments of f. A literal upon variable x +is either x or its negation x and a term is a conjunction of +literals. We often omit the conjunction symbols when writing +terms, for instance we may shorten a ∧ c into a c. We define +the empty term t∅ as the term over zero literal. The empty +term verifies t∧t∅ = t for every term t. Given ℓ ∈ {x, x}, we +denote by f|ℓ the Boolean function over var(f) \ {x} whose +satisfying assignments coincide with that of f ∧ℓ. We use the +usual symbols ∧, ∨, ¬, |= to denote conjunction, disjunction, +negation, and entailment. Given a set S of terms, max(S, |=) +denotes the subset of terms of S that do not entail another +term in S. An implicant of a Boolean function f is a term t +whose satisfying assignments also satisfy f, i.e., t |= f. An +implicant t is prime when no term t−ℓ obtained by removing +a literal ℓ from t is an implicant of f. +2.1 +Compilation Languages +Compilation languages are often seen as classes of circuits. +Let PS be a countable set of propositional variables. +A +circuit in negation normal form (NNF) is a directed acyclic +graph (DAG) whose leaves are labelled with 0 (false), 1 +(true), or a literal built upon x ∈ PS, and whose internal +nodes are labelled with ∧ or ∨ connectives; we call them ∧- +nodes and ∨-nodes. An NNF circuit computes a Boolean +function over the variables appearing in it. +For v a node +of an NNF circuit Σ, var(v) denotes the set of variables +labelling leaves under v in Σ and Σv denotes the subcir- +cuit of Σ rooted at v. The language of decomposable NNF +(DNNF) contains the NNF circuits where ∧-nodes are de- +composable, that is, the children v1, . . . , vm of every ∧-node +v are such that var(vi) ∩ var(vj) = ∅ for all i ̸= j. The lan- +guage of deterministic, decomposable NNF (d-DNNF) con- +tains the DNNF circuits Σ where ∨-nodes are deterministic, +that is, the children v1, . . . , vm of every ∨-node v are such +that Σvi ∧ Σvj is inconsistent for all i ̸= j. Finally, the +language of decision DNNF (dec-DNNF) is that of circuits +whose leaves are labelled with 0, 1, or a literal built upon + +x ∈ PS, and whose internal nodes are decision nodes and +∧-nodes. Whenever n is a decision node labelled by vari- +able x in a dec-DNNF circuit Σ, the circuit Σn given by +x +u +v +is viewed as a compact representation of the d- +DNNF circuit +∨ +∧ +∧ +x +x +u +v +(see Definition 2.6 and Figure +2 in [Darwiche and Marquis, 2002]). +Thus, a decision node n is labelled by a variable and has +two children: the 0-child (node u on the previous picture) and +the 1-child (node v on the previous picture). If n is labelled by +x and Σu (resp. Σv) represents the function f0 (resp. f1), then +Σn represents the function (x ∧ f0) ∨ (x ∧ f1). For instance, +Figure 1a gives a dec-DNNF circuit whose deepest decision +node computes (s ∧ 1) ∨ (s ∧ p). It is worth mentioning +that all Boolean functions on finitely many variables can be +represented in dec-DNNF, or indeed in any of its subsets like +FBDD, OBDD, and DT. +Let Σ be a dec-DNNF circuit. +The size of Σ, denoted +by |Σ| is its number of edges. +From a dec-DNNF cir- +cuit Σ, one can easily derive in polynomial time a dec- +DNNF circuit equivalent to Σ where every ∧-node has ex- +actly two children. Since it is computationally harmless, for +the sake of simplicity, our enumeration algorithms suppose +that the dec-DNNF circuits satisfy this condition, so that their +size is at most twice their number of nodes. In the same +vein, we suppose that our dec-DNNF circuits have been re- +duced, i.e., every node v in Σ such that Σv computes the +0 function reduces to a leaf labelled by 0. Testing the sat- +isfiability of a dec-DNNF circuit is feasible in linear time +[Darwiche and Marquis, 2002], so reducing a dec-DNNF cir- +cuit also is a polynomial-time operation. +2.2 +Enumeration Complexity +We now recall some enumeration complexity classes as de- +scribed in [Strozecki, 2019]. Let V be an alphabet and let +A be a binary predicate in V ∗ × V ∗. +Given an instance +x ∈ V ∗ (the input), A(x) (the set of solutions) denotes the set +of all y ∈ V ∗ such that A(x, y). The enumeration problem +Enum·A is the function mapping x to A(x). Enum·A is in the +class EnumP if for every y ∈ A(x), |y| is polynomial in |x|, +and if deciding whether y is in A(x) is in P. EnumP does +not capture the complexity of computing the set of solutions +A(x), it serves more as a counterpart of NP for enumeration +problems. +The model used for the enumeration of solutions is the ran- +dom access machine (RAM) model. See [Strozecki, 2019] +for details on why RAM have been chosen for this task. +A RAM solves Enum·A if, for all x, it returns a se- +quence y1, . . . , ym of pairwise distinct elements such that +{y1, . . . , ym} = A(x). Enum·A is in OutputP if there is +a RAM solving Enum·A in time O(poly(|x| + |A(x)|)) on +every input x. OutputP is a relevant enumeration class when +the whole set of solutions is explicitly asked for. For instance, +the dualization of a monotone CNF formula φ is the task of +generating a DNF formula equivalent to φ. Because of the +monotony condition on φ, the terms used in any smallest DNF +formula equivalent to φ are precisely its prime implicants. +Thus, the dualization problem boils down to enumerating all +the prime implicants of φ. +For other applications, computing only a fixed number of +solutions may be enough. A RAM solves Enum·A in in- +cremental time f(t)g(n) if on every x, it runs in time time +O(f(t)g(|x|)) and returns a sequence y1, . . . , yt of t pairwise +distinct elements of A(x) when t ≤ |A(x)|, and the whole set +A(x) when t > |A(x)|. Enum·A is in IncP if there is a RAM +that solves A in incremental time O(tanb) for some constants +a and b. IncP has a characterization that uses the function +problem AnotherSol·A which, given x and S ⊆ A(x), re- +turns y ∈ A(x) \ S when S ̸= A(x), and false otherwise. +Proposition 1 ([Strozecki, 2019]). A problem Enum·A in +EnumP is in IncP if and only if AnotherSol·A is in FP. +Note that OutputP is thought to be distinct from +IncP [Strozecki, 2019]. +3 +Enum·IP from dec-DNNF is in OutputP +Let us first consider the problem of enumerating the prime +implicants of a Boolean function f given as a dec-DNNF cir- +cuit Σ, for short the prime implicants of Σ. Let IP(Σ, t) be the +binary predicate representing the relation that t is a prime im- +plicant of Σ. Then IP(Σ) denotes the set of prime implicants +of Σ. We extend the notation IP(·) to any Boolean function f. +To be able to speak of prime implicants enumeration from cir- +cuits other than dec-DNNF ones we write “Enum.IP from L” +with L the language Σ belongs to. +We start with a couple of easy results. First of all, since +there is a linear-time procedure to verify that a term is an im- +plicant of a dec-DNNF circuit, there is a polynomial-time al- +gorithm to decide whether a given a term is a prime implicant +of a dec-DNNF circuits, thus: +Proposition 2. Enum·IP from dec-DNNF is in EnumP. +In addition, it is known that Enum·IP from OBDD is +in OutputP [Madre and Coudert, 1991], and it is almost +straightforward to extend this result to dec-DNNF. To make +it precise, let us briefly describe the output polynomial con- +struction of IP(Σ) for Σ, a dec-DNNF circuit. The construc- +tion is based on the three following, folklore propositions (for +the sake of completeness, a proof for each of them is nonethe- +less reported as a supplementary material). +Proposition 3. Let f and g be Boolean functions, then IP(f ∧ +g) = max({t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g)}, |=). Furthermore +if var(f) ∩ var(g) = ∅, then IP(f ∧ g) = {t ∧ t′ | t ∈ +IP(f), t′ ∈ IP(g)}. +Proposition 4. Let f a Boolean function, let x be a variable, +and let ℓ ∈ {x, x}. Consider t ∈ IP(f|ℓ). If t |= f|ℓ, then +t ∈ IP(f), otherwise t ∧ ℓ ∈ IP(f). +Proposition 5. Let f be a Boolean function and let x be a +variable. +IP(f) = {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} +∪ {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} +∪ IP(f|x ∧ f|x) + +h +e +e +∧ +∧ +b +p +s +b +s +p +1 +p +(a) A dec-DNNF circuit +h +v0 : +e +e +v1 : +∧ +∧ +b +v2 : +p +s +b +s +v3 : +p +1 +p +S3 = ∅ +S0 = {h e b p , h p s , e p s} +S2 = {b p , p s} +S1 = {e b p , p s} +(b) A path followed in MissingIP +h +v0 : +e +e +v1 : +∧ +∧ +b +v2 : +p +s +b +s +v3 : +p +1 +p +h e b s ∈ IP(Σv0) +s ∈ IP(Σv3) +e b s ∈ IP(Σv1) +b s ∈ IP(Σv2) +(c) Propagation of an implicant +Figure 1: Generation of a new prime implicant from a dec-DNNF circuit +Note that t ∈ IP(f|x) (resp. IP(f|x)) entails f|x (resp. f|x) +if and only if t is subsumed by some term in IP(f|x∧f|x). As +a consequence, from IP(f|x) and IP(f|x), one can construct +IP(f|x∧f|x) in polynomial time thanks to Proposition 3 and +we use it to derive IP(f) thanks to Proposition 5. +We +also +have +that +(see +the +algorithm +for +condi- +tioning +a +prime +implicant +representation +provided +in +[Darwiche and Marquis, 2002]): +Proposition 6. Let f a Boolean function and let x be a vari- +able, then |IP(f)| ≥ max(|IP(f|x)|, |IP(f|x)|). +Consider now a dec-DNNF circuit Σ and an internal node +v with two children u and w. If the sets IP(Σu) and IP(Σw) +are provided, then IP(Σv) is obtained in polynomial time us- +ing Proposition 3 if v is a decomposable ∧-gate, and using +Proposition 5 if v is a decision node. Furthermore, in both +cases, we have |IP(Σv)| ≥ max(|IP(Σu)|, |IP(Σw)|). These +observations lead to a simple algorithm that generates IP(Σ) +by computing the sets IP(Σv) for every node v of Σ consid- +ered in a bottom-up way. Since constructing the set of prime +implicants for any node given that of its children is tractable, +since this set is smaller than |IP(Σ)|, and since it is computed +only once, the algorithm runs in time O(poly(|Σ|+|IP(Σ)|)). +Thus, we get: +Proposition 7. Enum·IP from dec-DNNF is in OutputP. +Example 1. We give the construction of the sets of prime +implicants for the nodes v1, v2, v3 in the dec-DNNF circuit Σ +represented on Figure 1b. +v3: the sets of prime implicants of the children are IP(1) = +{t∅} and IP(p) = {p}. Using Proposition 5 we have that +s ∧ t∅ = s and Σv3|s = p, so s ∧ t∅ ̸|= Σv3|s showing +that s ∈ IP(Σv3). We also have that Σv3|s = 1, so +s p |= Σv3|s showing that s p ̸∈ IP(Σv3). Finally, we +have that IP(Σv3|s ∧ Σv3|s) = {p} by Proposition 3, so +IP(Σv3) = {s, p}. +v2: the sets of prime implicants of the children are +IP(p) = {p} and IP(Σv3) so we compute IP(Σv2) = +{b p, b s, b p, p s} +v1: the sets of prime implicants of the children are IP(p ∧ +s) = {p s} and IP(Σv2) so we compute IP(Σv1) = +{e b p, e b p, e b s, p s} +4 +Enum·IP from dec-DNNF is in IncP +We now investigate Enum·IP from dec-DNNF from the in- +cremental enumeration perspective. Based on Proposition 1, +we design a tractable algorithm AnotherIP for solving the +problem AnotherSol·IP, thus showing that Enum·IP from +dec-DNNF is in IncP. +4.1 +Solving the decision variant of AnotherSol·IP +We first consider the decision variant of AnotherSol·IP from +dec-DNNF: given a dec-DNNF circuit Σ and a set S ⊆ +IP(Σ), return false if and only if S +̸= IP(Σ). +Recall +from the discussion preceding Proposition 7 that there is a +bottom-up procedure for generating all prime implicants of +the dec-DNNF circuit Σ. To address the decision variant of +AnotherSol·IP on inputs Σ and S, a reverse, top-down search +is performed, assuming that S is IP(Σ) until finding a contra- +diction. +Before defining what a contradiction means in this setting, +a few notations are useful. For t a term and X a set of vari- +ables, tX denotes the restriction of t to variables in X. Note +that if X and var(t) are disjoint, then tX is the empty term t∅. +Proposition 8. Let Σ be a dec-DNNF circuit and let S ⊆ +IP(Σ). If the root of Σ is an ∧-node, let u and w be its chil- +dren and let Su = {tvar(Σu) | t ∈ S} and Sw = {tvar(Σw) | +t ∈ S}. Then Su ⊆ IP(Σu) and Sw ⊆ IP(Σw) hold, and +S = IP(Σ) iff Su = IP(Σu) and Sw = IP(Σw) +and S = {tu ∧ tw | tu ∈ Su, tw ∈ Sw}. +Proposition 9. Let Σ be a dec-DNNF circuit whose root is +a decision node labelled by x. Let u be its 0-child and w +be its 1-child. Given S ⊆ IP(Σ), let Su = {t | t ∧ x ∈ +S} ∪ (S ∩ IP(Σu)), Sw = {t | t ∧ x ∈ S} ∪ (S ∩ IP(Σw)) +and S′ = {t | t ∈ S, x ̸∈ var(t)}. Then Su ⊆ IP(Σu) and +Sw ⊆ IP(Σw) hold, and +S = IP(Σ) iff Su = IP(Σu) and Sw = IP(Σw) +and S′ = max({tu ∧ tw | tu ∈ Su, tw ∈ Sw}, |=). +Let v be the root of the dec-DNNF circuit Σ and let S ⊆ +IP(Σ). We say that we have a contradiction at node v when +(c1) S = ∅ while Σ is satisfiable, or +(c2) v is a decision node, Su = IP(Σu) and Sw = IP(Σw), +but S′ ̸= max({tu ∧ tw | tu ∈ Su, tw ∈ Sw}, |=), or + +Algorithm 1: MissingIP(Σ, S, P) +Promises: Σ is reduced, S ⊆ IP(Σ) +1 Let v be the root of Σ and let P ′ ← P ∪ (v) +2 if λ(v) = |S| then return false +3 if S = ∅ then +4 +if v is labelled by 0 then set λ(v) to 0, return false +5 +else return (GenerateIP(Σ), P ′) +6 end +7 if v is a ∧-node with children u and w then +8 +Build Su and Sw as in Proposition 8 +9 +r ← MissingIP(Σu, Su, P ′) +10 +if r ̸= false then return r +11 +r ← MissingIP(Σw, Sw, P ′) +12 +if r ̸= false then return r +13 +S∗ ← {tu ∧ tv | tu ∈ Su, tw ∈ Sw} +14 +if S ̸= S∗ then for any t ∈ S∗ \ S return (t, P ′) +15 else if v is a decision node with children u and w then +16 +Build Su, Sw, S′ as in Proposition 9 +17 +r ← MissingIP(Σu, Su, P ′) +18 +if r ̸= false then return r +19 +r ← MissingIP(Σw, Sw, P ′) +20 +if r ̸= false then return r +21 +S∗ ← max({tu ∧ tw | tu ∈ Su, tw ∈ Sw}, |=) +22 +if S∗ ̸= S′ then for any t ∈ S∗ \ S′ return (t, P ′) +23 end +24 Set λ(v) to |S| and return false +(c3) v is a decomposable ∧-node and Su = IP(Σu) and +Sw = IP(Σw) but S ̸= {tu ∧ tw | tu ∈ Su, tw ∈ Sw}. +A contradiction guarantees that S ̸= IP(Σ). +The contra- +diction (c1) is easy to check. Contradictions (c2) and (c3) +on the other hand require to show that Su = IP(Σu) and +Sw = IP(Σw). When v is an internal node, with children u +and w, if there is no contradiction (c1) at v, we use Propo- +sitions 8 and 9 to build from S two sets Su and Sw that we +recursively compare to IP(Σu) and IP(Σw). Either the re- +cursion ends under u or w on a contradiction, in which case +S ̸= IP(Σ), or it stops by itself (i.e., when reaching the +leaves of the circuit), which shows that Su = IP(Σu) and +Sw = IP(Σw), and then we can check whether there is con- +tradiction (c2) (resp. (c3)) at node v if it is a decision node +(resp. decomposable node). If there is none, then S = IP(Σ). +The procedure is given by Algorithm MissingIP. The +inputs are a dec-DNNF circuit Σ, a set S ⊆ IP(Σ) and a path +P in Σ (which will be useful later). A function λ mapping +the nodes of Σ to integers is used for memoization purposes. +Initially λ(v) = −1 for every node v, but λ(v) may be as- +signed a non-negative value at some point. More precisely, +the first time a call MissingIP(Σv, S, P) returns false, we +learn that S = IP(Σv) and set λ(v) to |S|. Then for each +later call MissingIP(Σv, S′, P ′) with S′ ⊆ IP(Σv), we +check whether S′ = IP(Σv) by verifying that λ(v) = |S′|. +Proposition 10. Given a reduced dec-DNNF circuit Σ and +S ⊆ IP(Σ), MissingIP(Σ, S, ∅) runs in time O(poly(|S|+ +|Σ|)), and it returns false if and only if S = IP(Σ). +Algorithm 2: GenerateIP(Σ) +Promise: Σ is satisfiable +1 Find a satisfying assignment a of Σ +2 Let t = � +a(x)=1 x ∧ � +a(x)=0 x +3 while there is ℓ ∈ t such that t − ℓ |= Σ do +4 +Remove ℓ from t +5 end +6 Return t +4.2 +Augmenting an incomplete subset of IP(Σ) +We build upon MissingIP so that, when S ̸= IP(Σ), we +also return a prime implicant in IP(Σ) \ S. The idea is to +use the path P to keep track of the ancestor nodes that were +visited before reaching a contradiction and to use P to con- +struct a prime implicant in IP(Σ) \ S. As an example, con- +sider calling MissingIP(Σ, S0, ∅) with Σ the dec-DNNF +circuit of Figure 1a and S0 = {h eb p, h p s, e p s} a set of +prime implicants of Σ. Figure 1b shows a scenario when +MissingIP(Σ, S0, ∅) calls MissingIP(Σv1, S1, (v0)), +which calls in turnMissingIP(Σv2, S2, (v0, v1)), which fi- +nally calls MissingIP(Σv3, S3, (v0, v1, v2)). Since S3 = ∅ +and Σv3 is reduced and different from 0, the algorithm has +reached a contradiction (c1) at node v3 and has not returned +false, thus indicating that S0 ̸= IP(Σ). MissingIP has fol- +lowed the path P = (v0, v1, v2, v3) to reach that contradiction +and has kept it in memory. This path P can then be used to +generate a prime implicant in IP(Σ) \ S0. First MissingIP +returns the path P to v3 as well as a prime implicant of Σv3, +say it is s. Then we construct a prime implicant of Σv2 upon +s, here since v3 is the 0-child of v2 and since s does not entail +the 1-child of v2 we obtain b s ∈ IP(Σv2). Then we con- +struct a prime implicant of Σv1 upon b s, here since since v2 +is the 0-child of v1 and since b s does not entail the 1-child of +v1 we obtain e b s ∈ IP(Σv1). Repeating the step one more +time leads to h e b s ∈ IP(Σv0) = IP(Σ). The procedure +is illustrated in Figure 1c. In this example, for generating a +new prime implicant of Σ, we have created t ∈ Σv3 \ S3 and +augmented it using Proposition 4 as we travelled backwards +along P. We say that we have propagated t along the path P. +Accordingly, the algorithm AnotherIP to generate +a new prime implicant breaks into two steps. +First +MissingIP(Σ, S, P) searches for a contradiction. It re- +turns false if S = IP(Σ) or a pair (t, P) with P the path +followed to reach a node v where a contradiction has been +found (like v3 in the example), and t a prime of Σv that could +not be derived from S. The procedure GenerateIP is used +to generate t. GenerateIP runs in polynomial time thanks +to linear-time implicant check on dec-DNNF circuits. Finally +PropagateIP is called to propagate t along the path P. +The +next +proposition +shows +the +correctness +of +AnotherIP: +Proposition 11. Let Σ be a reduced dec-DNNF circuit and let +S ⊆ IP(Σ). AnotherIP(Σ, S) runs in time O(poly(|S| + +|Σ|)). It returns false if S = IP(Σ), otherwise it returns a +prime implicant of Σ that does not belong to in S. +On this basis, the existence of a polynomial incremental + +Algorithm 3: Propagate(Σ, t, P = (v0, . . . , vi)) +Promise: Σ is reduced, its root is v0, P is a path in Σ +1 if |P| = 1 then return t +2 if vi−1 is a ∧-node with children u and w then +3 +if vi = u then t′ ← GenerateIP(Σw) +4 +if vi = w then t′ ← GenerateIP(Σu) +5 else if vi−1 is a decision node for variable x with +0-child u and 1-child w then +6 +if vi = u then +7 +if t |= Σw then t′ ← t∅ else t′ ← x +8 +else +9 +if t |= Σu then t′ ← t∅ else t′ ← x +10 end +11 Propagate(Σ, t ∧ t′, (v0, . . . , vi−1)) +Algorithm 4: AnotherIP(Σ, S) +Promise: Σ is reduced, S ⊆ IP(Σ) +1 r ← MissingIP∗(Σ, S, ∅) +2 if r = false then return false +3 else if r = (t, P) then return Propagate(Σ, t, P) +time enumeration of prime implicants for dec-DNNF circuits +can be easily established: +Proposition 12. Enum·IP from dec-DNNF is in IncP. +5 +Enumerating Specific Prime Implicants +For some applications, enumerating all prime implicants of +f makes sense, even though there can be exponentially many. +We have already mentioned the dualization of monotone CNF +formulae as an example. In this section, we describe two +problems that ask for generating only specific prime impli- +cants, representing respectively subset-minimal abductive ex- +planations and sufficient reasons. +To illustrate the two notions we use the function f com- +puted by the dec-DNNF circuit of Figure 1a as a toy example. +f encodes a very incomplete characterization of human-like +creatures in Tolkien’s Middle Earth based on four physical at- +tributes: presence of beard and facial hair (b), small size (s), +human-like skin (h), pointy ears (p), plus the indication of +whether the creature is enrolled in the armies of evil (e). We +imagine that there are only seven possible creatures: hobbits +(h b p s e), elves (h b p s e), dwarfs (h b p s e), men and women +(h∗ps ∗),1 ents (h∗p s e), orcs (h b p∗e) and trolls (h b p s e). +The satisfying assignments of f describe these creatures. Its +prime implicants are the smallest combinations of attributes +which guarantee the existence of a creature in our Middle +Earth. +5.1 +Abductive Explanations +Abductive +explanations +(see +e.g., +[Selman and Levesque, 1990; +Eiter and Gottlob, 1995]) +1∗ denotes that both choices are possible for the variable, typi- +cally here humans may fight for evil, humans and ents may or may +not have beards, and orcs have a wide range of size. +can be defined as follows: +Definition 1 (Abductive explanation). Given a Boolean +function f over variables X, a subset H ⊆ X, and a term +m on X \ H, an abductive explanation is a term t on H such +that f ∧ t is satisfiable and f ∧ t |= m. +The abduction problem asks whether an abductive explana- +tion t exists for the input (f, H, m). +Example 2. Consider our toy example. We look for combi- +nations of physical attributes that guarantee that the creature +is evil. This is an abduction problem with H = {h, b, p, s} +and m = e. For instance the term h ∧ p is an abductive ex- +planation because there exist creatures with pointy ears and a +skin that is not human-like, and all of them are evil (in this +case only the orcs fit this description). +It is easy to see that an abductive explanation t is in fact an +implicant of ¬f ∨m with the conditions that f ∧t is satisfiable +and that t is restricted to variables in H (the abducibles). Fur- +thermore, since abduction is not a truth-preserving form of in- +ference, one is often interested in generating subset-minimal +abductive explanations only (i.e., the logically weakest ab- +ductive explanations); they correspond to the prime impli- +cants of ¬f ∨m such that f ∧t is satisfiable and t is restricted +to variables in H. +Obviously enough, the abduction problem we focus on (the +existence of an abductive explanation) is the same, would +we consider subset-minimal abductive explanations or not. +Indeed, deciding whether an abductive explanation exists is +equivalent to deciding whether a subset-minimal abductive +explanation exists. +Unfortunately, the condition that only +variables in H are allowed in abductive explanations is al- +ready too demanding from an enumeration perspective. +Proposition 13. Unless P = NP, there is no polynomial-time +algorithm which, given an OBDD circuit or a decision tree +computing a function f over X and a set Y ⊆ X, decides +whether f has an implicant t with var(t) ⊆ Y . +5.2 +Sufficient Reasons +The notion of sufficient reason2 [Darwiche and Hirth, 2020] +(aka prime implicant explanation [Shih et al., 2018b]) is de- +fined as follows: +Definition 2 (Sufficient reason). Given a Boolean function +f, let a be any assignment to a superset of var(f). A suf- +ficient reason for a is a prime implicant t of f (resp. ¬f) +such that a satisfies t, provided that a satisfies f (resp. ¬f). +The set of all sufficient reasons for a given f is denoted by +SR(f, a) (resp. SR(¬f, a)) when a satisfies f (resp. ¬f). +Example 3. Consider again our toy example. There is no +creature which is small, has human-like skin, pointy ears, no +facial hair, and is evil. Finding the reasons of why such a +creature cannot exist, means finding sufficient reasons for the +assignment a defined by a(h) = a(p) = a(s) = a(e) = 1 +2This concept is also referred to as “abductive explanations” +[Ignatiev et al., 2019; Ignatiev et al., 2020]; in the following, we +stick to “sufficient reason” to avoid any confusion with the (distinct) +concept of abductive explanations as discussed in the previous sec- +tion. + +and a(b) = 0 given ¬f. In this case h p e ∈ SR(¬f, a) +explains why such a creature cannot exist: there are no crea- +tures that are evil and have both human-like skin and pointy +ears, but there are such creatures that are non-evil (hobbits +and elves), and there are evil creatures that have pointy ears +(orcs) or human-like skin (men). There are other sufficient +reasons for a given ¬f, for instance h s e ∈ SR(¬f, a). +We define the problem Enum·SR similarly to Enum·IP. A +couple of results about the complexity of computing sufficient +reasons have been pointed out for the past few years. Obvi- +ously enough, when no assumption is made on the represen- +tation of f, computing a single sufficient reason for an assign- +ment a is already NP-hard (for pretty much the same reasons +as for the prime implicant case, i.e., f is valid iff for any a, +the unique sufficient reason for a given f is the empty term). +Furthermore, the number of sufficient reasons for an assign- +ment a given f can be exponential in the number of variables +even when f is represented in DT [Audemard et al., 2021]. +Contrary to abductive explanations, it is computationally easy +to generate a single sufficient reason from SR(Σ, a) when +Σ is an OBDD circuit or a decision tree representing f. A +greedy algorithm can be used to this end: if a satisfies Σ +(resp. ¬Σ), then start with the canonical term having a as +its unique satisfying assignment and remove literals from +this term while ensuring that it still is an implicant of Σ +(resp. ¬Σ), until no more literal can be removed. In addi- +tion, when Σ is in DT, we can generate in polynomial time +a monotone CNF formula Ψ such that IP(Ψ) = SR(Σ, a) +(see [Darwiche and Marquis, 2021] for details), and then take +advantage of a quasi-polynomial time algorithm for enumer- +ating the elements of IP(Ψ) [Gurvich and Khachiyan, 1999]. +Contrastingly, deciding whether a preset number of sufficient +reasons for a given a exists is intractable (NP-hard), even +when the Boolean function f is monotone (see Theorem 3 +in [Marques-Silva et al., 2021]). +In the following, we complete those results by provid- +ing evidence that Enum·SR from any language among dec- +DNNF, OBDD, or DT is a difficult problem, despite the fact +that those languages are quite convenient for many reasoning +tasks [Darwiche and Marquis, 2002; Koriche et al., 2013]. +Let us first give an inductive computation of SR(Σ, a) sim- +ilar to that of IP(Σ). +Proposition 14. Let f and g be Boolean functions with +var(f) ∩ var(g) = ∅ and let a be a truth assignment to a +superset of var(f) ∪ var(g), then SR(f ∧ g, a) = {t ∧ t′ | +t ∈ SR(f, a), t′ ∈ SR(g, a)}. +Proposition 15. Let f be a Boolean function, let a be a truth +assignment to a superset of var(f) and let x ∈ var(f). If a +satisfies the literal ℓ on variable x then +SR(f, a) = {t ∧ ℓ | t ∈ SR(f|ℓ, a), t ̸|= f|ℓ} +∪ SR(f|x ∧ f|x, a). +By Proposition 6, |IP(f)| ≥ max(|IP(f|x)|, |IP(f|x)|). In a +sense this means that using IP(f|x) and IP(f|x) to generate +IP(f) is not a waste of resources since all these implicants +are kept in some form through IP(f). This led to our out- +put polynomial procedure to generate IP(f) for OBDD and +more generally for dec-DNNF circuits. On the other hand, it +is not guaranteed that SR(f, a) is larger than SR(f|x, a) and +SR(f|x, a) so there is no straightforward adaptation of this +procedure from Enum·IP to Enum·SR. +Example 4. Let Σ be the dec-DNNF circuit of Figure 1a. +Consider the dec-DNNF circuit Σv1 rooted at node v1, as +spotted in Figure 1b. The assignment a to {b, e, p, s} de- +fined by a(b) += +a(e) += +1 and a(p) += +a(s) += +0 +satisfies Σv1. +Recall that the set IP(Σv1) has been con- +structed in Example 1 and observe that SR(Σv1, a) += +{p s}. Now the 0-child of v1 is v2 and looking at the set +IP(Σv2) constructed in Example 1, we see that SR(Σv2, a) = +{p s, b p}. Since Σv2 = Σv1|e, we have that |SR(Σv1, a)| < +max(|SR(Σv1|e, a)|, |SR(Σv1|e, a)|). +Actually, we give evidence that enumerating sufficient rea- +sons from dec-DNNF, and even from OBDD or DT, is not in +OutputP by reducing to it the problem of enumerating the +minimal transversals of a hypergraph, a well-known problem +whose membership to OutputP is a long-standing question. +Formally: +Proposition 16. If Enum·SR from OBDD is in OutputP or +Enum·SR from DT is in OutputP, then enumerating the min- +imal transversals of a hypergraph is in OutputP. +6 +Conclusion +Most applications of prime implicants for Boolean function +analysis use only a fraction of the many prime implicants +a Boolean function may have. +Especially, in the context +of logic-based abduction, subset-minimal assumptions to be +added to the available background knowledge in order to be +able to derive some given manifestations are looked for; in +the propositional case, they correspond to specific prime im- +plicants. Furthermore, in an eXplainable AI perspective, spe- +cific prime implicants known as sufficient reasons are used to +explain the predictions of machine learning algorithms. +In our work, we have studied the enumeration of general +and specific prime implicants of Boolean functions repre- +sented as dec-DNNF circuits. It was known that these circuits +enable efficient reasoning on Boolean functions. Our results +show that when it comes to prime implicants enumeration, +dec-DNNF circuits have benefits as well as limitations. Our +take-home message is that, while dec-DNNF circuits enable +enumerating general prime implicants in incremental polyno- +mial time, there are strong pieces of evidence against the exis- +tence of any output-polynomial time procedure for enumerat- +ing specific prime implicants from dec-DNNF circuits. More +precisely, if a procedure for enumerating subset-minimal ab- +ductive explanations were to exist, then P = NP would fol- +low. Similarly, if there were an output-polynomial time al- +gorithm for enumerating sufficient reasons from dec-DNNF +circuits, then the enumeration of the minimal transversals of +a hypergraphwould be in OutputP. Though this is considered +unlikely in enumeration complexity, we think that proving a +stronger statement would be a valuable contribution. We let +this task open for future research. + +Acknowledgments +Many thanks to the anonymous reviewers for their comments +and insights. +This work has benefited from the supports +of the PING/ACK project (ANR-18-CE40-0011) and of the +AI Chair EXPEKCTATION (ANR-19-CHIA-0005-01)of the +French National Research Agency. It was also partially sup- +ported by TAILOR, a project funded by EU Horizon 2020 +research and innovation programme under GA No 952215. +References +[Audemard et al., 2021] G. Audemard, S. Bellart, Loue- +nas Bounia, F. Koriche, J.-M. Lagniez, and P. Marquis. +On the explanatory power of decision trees. +CoRR, +abs/2108.05266, 2021. +[Bryant, 1986] R. E. Bryant. +Graph-based algorithms for +Boolean function manipulation. +IEEE Transactions on +Computers, C-35(8):677–692, 1986. +[Chandra and Markowsky, 1978] A.K. +Chandra +and +G. Markowsky. +On the number of prime implicants. +Discrete Mathematics, 24:7–11, 1978. +[Darwiche and Hirth, 2020] A. Darwiche and A. Hirth. 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Direct +from +the +fact +that +dec-DNNF +is +a +sublanguage +of +deterministic +DNNF +(d-DNNF) +and +d-DNNF +supports +polynomial +time +implicant +check [Darwiche and Marquis, 2002]. +Proposition 3. Let f and g be Boolean functions, then IP(f ∧ +g) = max({t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g)}, |=). Furthermore +if var(f) ∩ var(g) = ∅, then IP(f ∧ g) = {t ∧ t′ | t ∈ +IP(f), t′ ∈ IP(g)}. +Proof. A proof of IP(f ∧ g) = max({t ∧ t′ | t ∈ IP(f), t′ ∈ +IP(g)}, |=) can be found e.g., in [Marquis, 1993]. +In the case where var(f) ∩ var(g) = ∅, the terms in IP(f) +contain only variables from var(f) and the terms in IP(g) +contain only variables from var(g). +We denote lit(f) = +{x, x | x ∈ var(f)}. Let tf, t′ +f ∈ IP(f) and tg, t′ +g ∈ IP(g) +such that tf ∧ tg |= t′ +f ∧ t′ +g. Looking at terms as sets of liter- +als this means that t′ +f ∪ t′ +g ⊆ tf ∪ tg. But then t′ +f ⊆ tf since +(tf ∪tg)∩lit(f) = tf and (t′ +f ∪t′ +g)∩lit(f) = t′ +f. This means +that tf |= t′ +f and therefore tf = t′ +f. A similar argument gives +that tg = t′ +g. This shows that when var(f) ∩ var(g) = ∅, +IP(f ∧ g) = max({t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g)}, |=) = +{t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g)}. +Proposition 4. Let f a Boolean function, let x be a variable, +and let ℓ ∈ {x, x}. Consider t ∈ IP(f|ℓ). If t |= f|ℓ then +t ∈ IP(f), otherwise t ∧ ℓ ∈ IP(f). +Proof. Suppose t |= f|ℓ. Then t is an implicant of f since +t |= (f|x∧f|x) |= ((x∧f|x)∨(x∧f|x)) ≡ f. To prove that +it is prime let t′ be a strict subterm of t and assume t′ |= f. +We have x ̸∈ var(t) since x ̸∈ var(f|ℓ), so t′|ℓ = t′. If +t′ |= f then t′ = t′|ℓ |= f|ℓ and t is not a prime implicant of +f|ℓ, a contradiction. +Suppose t ̸|= f|ℓ. Then t ∧ ℓ is an implicant of f since +t ∧ ℓ |= (ℓ ∧ f|ℓ) |= ((x ∧ f|x) ∨ (x ∧ f|x)) ≡ f. To prove +that it is prime, let t′ be a strict subterm of t ∧ ℓ and assume +t′ |= f. If ℓ ̸∈ t′ then t′ = t′|ℓ |= f|ℓ and t is not a prime +implicant of f|ℓ, a contradiction. If however ℓ ∈ t′, then write +t′ = t′′ ∧ ℓ and observe that t′′ = t′|ℓ |= f|ℓ, so t is not a +prime implicant of f|ℓ, another contradiction. +Proposition 5. Let f be a Boolean function and let x be a +variable. +IP(f) = {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} +∪ {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} +∪ IP(f|x ∧ f|x) +Proof. We derive {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} ∪ {t ∧ x | +t ∈ IP(f|x), t ̸|= f|x} ⊆ IP(f) from Proposition 4. +Now we show that IP(f|x ∧ f|x) ⊆ IP(f). +Let t ∈ +IP(f|x ∧ f|x), then t |= f|x and t |= f|x. +Since x ̸∈ +var(f|x)∪var(f|x) we have that x ̸∈ var(t). First we prove +that t is an implicant of f. If we had t ̸|= f then t|x ̸|= f|x or +t|x ̸|= f|x would hold, but t|x = t|x = t so t |= f. Now to +prove that it is a prime implicant. Let ℓ ∈ t and let t′ be the +term t deprived from ℓ. If t′ |= f were to hold then so would +t′|x |= f|x and t′|x |= f|x. But since t′|x = t′|x = t′, this +would mean that t′ |= f|x ∧ f|x and therefore t would not be +a prime implicant of f|x ∧ f|x, a contradiction. This shows +that IP(f|x ∧ f|x) ⊆ IP(f). +We have established that +IP(f) ⊇ {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} +∪ {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} +∪ IP(f|x ∧ f|x) +and now we show the reverse inclusion. Let t ∈ IP(f). As- +sume t = t0 ∧ x. t |= f implies that t|x |= f|x, or in other +words, that t0 |= f|x. If t0 was not a prime implicant of f|x, +that is, if there was t′ +0 ⊂ t0 such that t′ +0 |= f|x, then we +would also have t′ +0 ∧ x |= f and therefore t would not be a +prime implicant of f. So t0 ∈ IP(f|x). Now if t0 |= f|x +then we would have t0 |= f|x ∧ f|x |= f, so t would not +be a prime implicant of f, a contradiction. This shows that +if x is in t, then t ∈ {t0 ∧ x | t0 ∈ IP(f|x), t0 ̸|= f|x}. A +symmetrical proof gives that if x is in t, then t ∈ {t1 ∧ x | +t1 ∈ IP(f|x), t1 ̸|= f|x}. +Finally if neither x nor x is in t, then t = t|x |= f|x and +t = t|x |= f|x, and therefore t |= f|x∧f|x. Now if t was not +a prime implicant of f|x∧f|x then there would be some t′ ⊂ +t in IP(f|x ∧ f|x). Since IP(f|x ∧ f|x) ⊆ IP(f), this would +mean that t is not a prime implicant of f, a contradiction. So +t ∈ IP(f|x ∧ f|x). +We have established that +IP(f) ⊆ {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} +∪ {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} +∪ IP(f|x ∧ f|x) +thus finishing the proof. +Proposition 6. Let f a Boolean function and let x be a vari- +able, then |IP(f)| ≥ max(|IP(f|x)|, |IP(f|x)|). +Proof. Let +ℓ +∈ +{x, x}. +It +is +shown +in [Darwiche and Marquis, 2002] that IP(f|ℓ) = max({t|ℓ | +t ∈ IP(f)}, |=). So |IP(f|ℓ)| = | max({t|ℓ | t ∈ IP(f)}, |= +)| ≤ |{t|ℓ | t ∈ IP(f)}| = |IP(f)|. +Proposition 7. Enum·IP from dec-DNNF is in OutputP. + +Proof. Given a dec-DNNF circuit Σ, we construct IP(Σ) by +visiting every node v of Σ in a bottom-up order while com- +puting IP(Σv). We start from the leaves. If v is labelled +by a literal ℓ then IP(Σv) = {ℓ}, if it is labelled by 0 then +IP(Σv) = ∅, and if it is labelled by 1 then IP(Σv) = {t∅} +where t∅ is the term containing no literal. +Now let v be an internal node of Σ and let u and w be +its children. Since we visit the nodes in depth-first order, +IP(Σu) and IP(Σw) have already been computed. If v is +a decomposable ∧-node then using Proposition 3 we com- +pute IP(Σv) = {tu ∧ tw | tu ∈ IP(Σu), tw ∈ IP(Σw)} in +time O(|IP(Σu)| × |IP(Σw)|) = O(|IP(Σ)|2). Observe that +|IP(Σv)| ≥ max(|IP(Σu)|, |IP(Σw)|). +If v is a decision node for the variable x whose 0- and 1- +children are u and w, respectively, then we compute IP(Σv) +from IP(Σu) and IP(Σw) using Proposition 5. Since dec- +DNNFs support linear-time implicant check, {t ∧ x | t ∈ +IP(Σu), t ̸|= Σw} and {t ∧ x | t ∈ IP(Σw), t ̸|= Σu} are +build in time polynomial in |Σ| + |IP(Σu)| + |IP(Σw)|. As +for IP(Σu∧Σw), we build it in time polynomial in |IP(Σu)|+ +|IP(Σw)| using Proposition 3. Observe that, by Proposition 6, +there is again |IP(Σv)| ≥ max(|IP(Σu)|, |IP(Σw)|). +When we reach the root node r we compute IP(Σr) = +IP(Σ). Since for every node v with children u and w we +have that |IP(Σv)| ≥ max(|IP(Σu)|, |IP(Σw)|), we also have +that |IP(Σv)| ≤ |IP(Σ)| for every v. We build IP(Σv) only +once and the time spent on each node v to build IP(Σv) given +IP(Σu) and IP(Σw) is polynomial in |Σ|+|IP(Σ)|. Summing +over all nodes we get that the time needed to build IP(Σ) is +also polynomial in |Σ| + |IP(Σ)|. +Proposition 8. Let Σ be a dec-DNNF circuit and let S ⊆ +IP(Σ). If the root of Σ is an ∧-node, let u and w be its chil- +dren and let Su := {tvar(Σu) | t ∈ S} and Sw := {tvar(Σw) | +t ∈ S}. Then Su ⊆ IP(Σu) and Sw ⊆ IP(Σw) hold, and +S = IP(Σ) iff Su = IP(Σu) and Sw = IP(Σw) +and S = {tu ∧ tv | tu ∈ Su, tv ∈ Sv}. +Proof. If S = IP(Σ) then by Proposition 3 S = {tu ∧ tv | +tu ∈ IP(Σu), tv ∈ IP(Σv)} so IP(Σu) = {tvar(Σu) | t ∈ +S} = Su and IP(Σv) = {tvar(Σv) | t ∈ S} = Sv and thus +S = {tu ∧ tv | tu ∈ Su, tv ∈ Sv}. +If S ̸= IP(Σ), let t∗ ∈ IP(Σ) \ S and let t∗ +u = t∗ +var(Σu) and +t∗ +v = t∗ +var(Σv). By Proposition 3, t∗ +u (resp. t∗ +v) is in IP(Σu) +(resp. IP(Σv)), so either t∗ +u ̸∈ Su or t∗ +v ̸∈ Svand we are done, +or t∗ +u ∈ Su and t∗ +v ∈ Sv in which case {tu∧tv | tu ∈ Su, tv ∈ +Sv} ̸= S since t∗ +u ∧ t∗ +v is in {tu ∧ tv | tu ∈ Su, tv ∈ Sv} but +not in S. +Proposition 9. Let Σ be a dec-DNNF circuit whose root is +a decision node labelled by x and whose 0- and 1-child are +u and w. Given S ⊆ IP(Σ), let Su = {t | t ∧ x ∈ S} ∪ +(S ∩ IP(Σu)), Sw = {t | t ∧ x ∈ S} ∪ (S ∩ IP(Σw)), +S′ = {t | t ∈ S, var(x) ̸∈ var(t)}. Then Su ⊆ IP(Σu) and +Sw ⊆ IP(Σw) hold, and +S = IP(Σ) iff Su = IP(Σu) +and Sw = IP(Σw) +and S′ = max({tu ∧ tw | tu ∈ Su, tw ∈ Sw}, |=). +Algorithm 5: GenerateIP(Σ) +Promise: Σ is satisfiable +1 Find a satisfying assignment a of Σ +2 Let t be the corresponding term: +t = � +a(x)=1 x ∧ � +a(x)=0 x +3 while there is ℓ ∈ t such that t − ℓ |= Σ do +4 +Remove ℓ from t +5 end +6 return t +Proof. For convenience we denote S∗ = max({tu ∧ tw | +tu ∈ Su, tw ∈ Sw}, |=). +First we prove that Sw ⊆ IP(Σw) (the proof that Su ⊆ +IP(Σu) is analogous). Clearly S ∩ IP(Σw) ⊆ IP(Σw) so +we just need to show that {t | t ∧ x ∈ S} ⊆ IP(Σw). Let +t ∧ x be in S, then t ∧ x |= Σ. t is an implicant of Σw since +t ≡ (t ∧ x)|x |= Σ|x ≡ Σw. Now if there exists t′ ̸= t +such that t |= t′ |= Σw then t ∧ x |= t′ ∧ x |= Σ holds, and +therefore t ∧ x is not a prime implicant of Σ, a contradiction. +So {t | t ∧ x ∈ S} ⊆ IP(Σw). +Second we prove that Sw ̸= IP(Σw) implies S ̸= IP(Σ) +(the proof is similar for Su ̸= IP(Σu)). Assume there ex- +ists t ∈ IP(Σw) \ Sw. If t |= Σu then t is in IP(Σ) by +Proposition 4. But t cannot be in S for otherwise it would be +in S ∩ IP(Σw) ⊆ Sw. This shows that S ̸= IP(Σ) in this +case. If however t ̸|= Σu then t ∧ x is in IP(Σ) by Proposi- +tion 4. But t ∧ x cannot be in S for otherwise t would be in +{τ | τ ∧ x ∈ S} ⊆ Sw. So again S ̸= IP(Σ). +Now we prove that (S′ ̸= S∗) ⇒ (S ̸= IP(Σ)). We +may assume that Su = IP(Σu) and Sw = IP(Σw), otherwise +S ̸= IP(Σ) holds regardless of S′ = S∗. Since Σu ≡ Σ|x +and Σw = Σ|x we have that S∗ = max({tu ∧ tw | tu ∈ +IP(Σ|x), tw ∈ IP(Σ|x)}, |=) = IP(Σ|x ∧ Σ|x) by Proposi- +tion 3. Now S = S′ ∪ {t | t ∈ S, x ∈ t} ∪ {t | t ∈ S, x ∈ t} +so, by Proposition 5, if S = IP(Σ) then S′ corresponds to the +set IP(Σ|x ∧ Σ|x). So +(S′ ̸= S∗) ⇒ (S′ ̸= IP(Σ|x ∧ Σ|x)) ⇒ (S ̸= IP(Σ)) +Now for the other direction, assume there exists t ∈ +IP(Σ) \ S. First suppose that t = t′ ∧ x. On the one hand t′ +is in IP(Σ|x) = IP(Σw). On the other hand t′ is clearly not +in {τ | τ ∧ x ∈ S}, and since it is not an implicant of Σ, it is +not in S ∩ IP(Σw) either. This means that t′ ∈ IP(Σw) \ Sw +and therefore Sw ̸= IP(Σw). In the case where t = t′ ∧ x, a +similar proof gives that Su ̸= IP(Σu). It remains to consider +the situation where neither x nor x is in t. By Proposition 5, t +is contained in IP(Σ|x∧Σ|x). As before, we can assume that +Su = IP(Σu) and Sw = IP(Σw). We have already explained +that this assumption yields S∗ = IP(Σ|x ∧ Σ|x). Since t is +not in S and x ̸∈ t and x ̸∈ t, we have that t ̸∈ S′. So +t ∈ S∗ \ S′, and therefore S ̸= S∗. +Proposition 10. Given a reduced dec-DNNF circuit Σ and +S ⊆ IP(Σ), MissingIP(Σ, S, ∅) runs in time O(poly(|S|+ +|Σ|)), and it returns false if and only if S = IP(Σ). +Proof. Soundness: We prove soundness by induction on the +depth of Σ using Propositions 8 and 9. + +If Σ has depth 1 then it is a single node v labelled by 0, 1 or +a literal ℓ. The promise states that S ⊆ IP(Σ). If v is labelled +by 0, then S must be ∅ and the algorithm returns false at line +4. If v is is labelled by 1 then either S = {t∅} = IP(1) and the +algoritm returns false at line 24, or S = ∅ and the algorithm +returns (1, (v)) at line 5. Finally if v is labelled by ℓ, either +S = {ℓ} = IP(ℓ) and the algorithm returns false at line 24, or +S = ∅ and the algorithm returns (ℓ, (v)) at line 5. In all cases +the algorithm returns false if and only if S = IP(Σ), and it +sets λ(v) to |IP(Σv)| before returning false. +Now if Σ has depth more than 1, its root node v is ei- +ther a decomposable ∧-node or a decision node. Since Σ +is reduced, it cannot be unsatisfiable, so if S = ∅ the al- +gorithm returns (t, (v)) with t ∈ IP(Σ) at line 5. +From +now on we suppose that S ̸= ∅. If v is a decomposable +∧-node with children u and w. By Proposition 8, since we +are promised that S ⊆ IP(Σ), we have that IP(Σ) = S +if and only if IP(Σu) = Su and IP(Σw) = Sw and we +can construct S from Su and Sw as shown in Proposition 8 +(Su and Sw defined as in Proposition 8). +By induction +IP(Σu) ̸= Su or IP(Σw) ̸= Sw if and only if the output of +MissingIP(Σu, Su, ∗) or MissingIP(Σw, Sw, ∗) is dis- +tinct from false. So if IP(Σu) ̸= Su or IP(Σw) ̸= Sw, a +return statement occurs line 9 or 12. Otherwise, it possible +that IP(Σu) = Su or IP(Σw) = Sw but that S can not be +constructed from Su and Sw, then the return statement of line +14 is triggerd. So if S ̸= IP(Σ), then lines 8-14 return some- +thing that is not false. And if S = IP(Σ), then no return call +is triggered lines 8-14 and the algorithm returns false at line +24 after setting λ(v) to |S| = |IP(Σ)| = |IP(Σv)|. +If v is a decision node for variable x with 0-child u and +1-child w. By Proposition 9, since we are promised that S ⊆ +IP(Σ), we have that IP(Σ) = S if and only if IP(Σu) = Su +and IP(Σw) = Sw and S′ = S∗ with Su, Sw and S′ de- +fined as in Proposition 9 and S∗ defined line 21. By induction +IP(Σu) = Su and IP(Σw) = Sw if and only if the output of +MissingIP(Σu, Su, ∗) or MissingIP(Σw, Sw, ∗) is dis- +tinct from false. So if S ̸= IP(Σ), then lines 16-22 return +something that is not false. And if S = IP(Σ), then no return +call is triggered lines 16-22 and the algorithm returns false at +line 24 after setting λ(v) to |S| = |IP(Σ)| = |IP(Σv)|. +Running +time: +Consider +the +time +spent +in +MissingIP(Σ, S, P) before a return statement or a +recursive call is triggered. The procedure may end at line 2 +or 4 in O(1) time. It can also end line 5, in which case it +has to compute a prime implicant of Σ using GenerateIP, +which runs in time O(poly(|Σ|)). Now if the algorithm has +not returned lines 2, 4 or 5, most of the running time is spent +building sets of terms from S lines 8, 13, 16 and 21. Building +Su and Sw line 8 only requires projecting the terms in S onto +var(Σu) and var(Σw), which takes time O(|S|). Construct- +ing the set S∗ at line 13 takes O(|Su| × |Sw|) = O(|S|2) +time. At line 16, S′ can clearly be obtained in time O(|S|) +and Su and Sw are obtained in time O(poly(|S| + |Σ|)) +thanks to polynomial-time prime implicant check on dec- +DNNF circuits. Finally the set S∗ at line 21 is constructed +in O(|Su| × |Sw|) = O(|S|2) and compared to S′ in time +O(poly(|S|). +So before a return statement or a recursive +call is triggered, the algorithm spends O(poly(|S| + |Σ|) +time. One can observe that |Su|, |Sw| are fewer than |S|, so +for every node v in Σ, a call MissingIP(Σv, S′, ∗) takes +O(poly(|S| + |Σ|)) time before triggering a return statement +or a recursive call. Thanks to memoization – implemented +via λ – the O(poly(|S| + |Σ|)) time procedure is done only +once per node. So the total running time of the algorithm is +also in O(poly(|S| + |Σ|)). +Proposition 11. Let Σ be a reduced dec-DNNF circuit and let +S ⊆ IP(Σ). AnotherIP(Σ, S) runs in time O(poly(|S| + +|Σ|)). It returns false if S = IP(Σ), otherwise it returns a +prime implicant of Σ that does not belong to S. +Proof. Soundness. +First +AnotherIP(Σ, S) +calls +MissingIP(Σ, S, ∅). +Soundness of MissingIP has +been established in Proposition 10 so if S += +IP(Σ) +then +MissingIP(Σ, S, ∅) +returns +false +and +so +does +AnotherIP(Σ, S). +Now let us assume that MissingIP(Σ, S, ∅) has not +returned false but the pair (t, P) with P = (v0, . . . , vm) +a path from v0 (the root of Σ) to vm and t a term. +Use +the notation Pi = (v0, . . . , vi−1) for all 1 ≤ i ≤ m. +Then calling MissingIP(Σ, S, ∅) has triggered a se- +quence +of +recursive +calls +MissingIP(Σv1, S1, P1), +MissingIP(Σv2, S2, P2),. . . ,MissingIP(Σvm, Sm, Pm). +A contradiction has been found during the last step: +MissingIP(Σvm, Sm, Pm) ended line 5 for a contradiction +of type (c1), or line 20 for a contradiction of type (c2), +and returned (t, P) with t some term that we claim is in +IP(Σvm) \ Sm. +Claim 1. t ∈ IP(Σvm) \ Sm. +Proof. This is clear if MissingIP(Σvm, Sm, Pm) ends line +5. Now if it ends line 20, then vm is a decision node for x with +0-child u and 1-child w. The sets Su, Sw, S′ and S∗ have +been generated and that it has been shown that Su = IP(Σu) +and Sw = IP(Σw) (otherwise a return statement line 16 or +18 would have been triggered). So S∗ = IP(Σu ∧ Σw) = +IP(Σvm|x∧Σvm|x) by Proposition 3. We have t ∈ S∗ \S′ so +it is clear that x ̸∈ var(t). Furthermore S′ contains all terms +from Sm in which neither x nor x appears, so t ∈ S∗ \ S′ +really means that t ∈ S∗ \Sm = IP(Σvm|x∧Σvm|x)\Sm ⊆ +IP(Σvm) \ Sm. +Now +AnotherIP(Σ, S) +returns +the +result +Propagate(Σ, t, P). To prove that the output is a term in +IP(Σ) \ S, it is sufficient to show that, for every 1 ≤ i ≤ m, +if ti +∈ +IP(Σvi) \ Si then Propagate(Σ, ti, Pi) calls +Propagate(Σ, ti−1, Pi−1) with ti−1 ∈ IP(Σvi−1) \ Si−1. +The rest is an easy induction (with S0 = S and Σv0 = Σ). +Claim 2. +Let ti +∈ +IP(Σvi) \ Si +with +i +≥ +1 +then +Propagate(Σ, ti, Pi) +does +a +recursive +call +Propagate(Σ, ti−1, Pi−1) with ti−1 ∈ IP(Σvi−1) \ Si−1. +Proof. Propagate(Σ, ti, Pi) calls Propagate(Σ, ti ∧ +t′, Pi−1). +Let ti−1 = ti ∧ t′. +We need to show that it +is in IP(Σvi−1) \ Si−1. +First assume that vi−1 is a de- +composable ∧-node with children vi and w, then t′ is ob- +tained line 4 and clearly t′ ∈ IP(Σw). By Proposition 3, + +ti ∧ t′ ∈ IP(Σvi ∧ Σw) = IP(Σvi−1). +By construction +Si = {tvar(Σvi) | t ∈ Si−1}. If ti ∧ t′ was in Si−1 then +its restriction ti to var(Σvi) would be Si, a contradiction. So +ti ∧ t′ ̸∈ Si−1. +Now suppose vi−1 is a decision node for x with 0-child u +and 1-child w. Let vi be u (the case vi = w is analogous). +By construction Si = Su. t′ is obtained line 7 and, by Propo- +sition 4, ti ∧ t′ ∈ IP(Σvi−1). To prove that ti ∧ t′ ̸∈ Si−1, +first assume that ti |= Σw. Then t′ is the empty term t∅. So +ti ∧ t′ = ti and ti ∈ IP(Σvi−1). If ti was in Si−1 then we +would have ti ∈ Si−1 ∩ IP(Σvi) ⊆ Si, a contradiction. So +when ti |= Σw, we have ti ∧ t′ ∈ IP(Σvi−1) \ Si−1. Now if +ti ̸|= Σw, then ti ∧ t′ = ti ∧ x and ti ∧ x is not in Si−1 for +otherwise we would have ti ∈ {τ | τ ∧ x ∈ Si−1} ⊆ Si. So +again we have ti ∧ t′ ∈ IP(Σvi−1) \ Si−1. +Running time. It has already been proved in Proposition 10 +that Missing(Σ,S,∅) runs in time O(poly(|S| + |Σ|). As +for Propagate(Σ, t, P), |P| recursive calls are made and +the cost between two consecutive recursive calls is either one +call to GenerateIP line 3 or 4, or one implicant check line +7 or 9. An implicant test on a dec-DNNF takes linear time +and GenerateIP makes at most |var(Σ)| such tests, so it +runs in time O(poly(|Σ|)). Thus Propagate(Σ, t, P) runs +in time O(|P| × poly(|Σ|)) = O(poly(|Σ|)). +Proposition 12. Enum·IP from dec-DNNF is in IncP. +Proof. Using Proposition 11, k prime implicants of Σ can +be generated in time O(poly(k + |Σ|)) by simply calling +AnotherIP(Σ, S) k times, each time adding to S the new +prime implicant that has been computed. This shows that +Enum·IP from dec-DNNF is in IncP. +Proposition 13. Unless P = NP, there is no polynomial-time +algorithm which, given an OBDD circuit or a decision tree +computing a function f over X and a set Y ⊆ X, decides +whether f has an implicant t with var(t) ⊆ Y . +Proof. Let φ be a CNF formula with m clauses c1, . . . , cm. +Create m fresh variables z1, . . . , zm. +Let B1, . . . , Bm be +OBDD circuits respecting the same variable ordering and +computing c1, . . . , cm, respectively. These OBDD circuits +can be computed in polynomial time (and can even be chosen +in DT). Define now the OBDD circuits B(i) = (zi ∧ Bi) ∨ +(zi ∧ B(i+1)) for 1 ≤ i ≤ m, with B(m+1) = 1. B(1) is an +OBDD circuit on {z1, . . . , zm} ∪ var(φ) built in polynomial +time from φ and whose size is in O(|φ|). +Claim 3. An implicant t of B(1) such that var(t) ⊆ var(φ) +exists if and only if φ is satisfiable. +Proof. For the first direction assume the implicant exists. t +is an implicant of B(1) = (z1 ∧ B1) ∨ (z1 ∧ B(2)). Since +z1 ̸∈ var(t), we have t |= B1 ≡ c1 and t |= B(2). Following +the same line of reasoning with B(2) instead of B(1) we also +have that t |= B2 ≡ c2 and t |= B(3). And we repeat the +argument until reaching, t |= c1, t |= c2, . . . , t |= cm, t |= +B(m+1) = 1. So indeed t |= φ and then φ is satisfiable. +For the other direction assume φ is satisfiable. Then there +exists an implicant t of φ with var(t) ⊆ var(φ). Let a be a +truth assignment to var(φ) ∪ {z1, . . . , zm} that satisfies t. If +a(zi) = 1 for all i ∈ {1, . . ., m}, then B(1)|a ≡ B(2)|a ≡ +· · · ≡ B(m+1)|a ≡ 1. Otherwise let j be the smallest inte- +ger such that a(zj) = 0. Then B(1)|a ≡ B(2)|a ≡ · · · ≡ +B(j)|a ≡ Bj|a ≡ cj|a. Since t is an implicant of φ, we have +that t |= cj, so cj|a ≡ 1. Thus every assignment a that sat- +isfies t also satisfies B(1), in other words t is an implicant of +B(1). +So if the algorithm from the proposition statement exists, +we can run it on inputs B(1) and Y = var(φ) to decide in +polynomial time whether φ is satisfiable. +Finally +note +that +if +one +had +chosen +to +represent +B1, . . . , Bm as decision trees from DT (which is also fea- +sible in polynomial time), then B(1) would be an element of +DT. So the statement also holds for DT. +Proposition 14. Let f and g be Boolean functions with +var(f) ∩ var(g) = ∅ and let a be a truth assignment to a +superset of var(f) ∪ var(g), then SR(f ∧ g, a) = {t ∧ t′ | +t ∈ SR(f, a), t′ ∈ SR(g, a)}. +Proof. Comes from Proposition 3: +SR(f ∧ g, a) = {τ ∈ IP(f ∧ g) | a satisfies τ} += {t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g), a satisfies t ∧ t′} += {t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g), a satisfies both t and t′} += {t ∧ t′ | t ∈ SR(f, a), t′ ∈ SR(g, a)} +Proposition 15. Let f be a Boolean function, let a be a truth +assignment to a superset of var(f) and let x ∈ var(f). If a +satisfies the literal ℓ on variable x then +SR(f, a) = {t ∧ ℓ | t ∈ SR(f|ℓ, a), t ̸|= f|ℓ} +∪ SR(f|x ∧ f|x, a). +Proof. Comes from Proposition 5: +SR(f, a) ={t ∈ IP(f) | a satisfies t} +={t ∧ ℓ | t ∈ IP(f|ℓ), t ̸|= f|ℓ, a satisfies t} +∪ {t ∧ ℓ | t ∈ IP(f|ℓ), t ̸|= f|ℓ, a satisfies t} +∪ {t ∈ IP(f|x ∧ f|x) |, a satisfies t} +={t ∧ ℓ | t ∈ SR(f|ℓ, a), t ̸|= f|ℓ} +∪ SR(f|x ∧ f|x, a). +Proposition 16. If Enum·SR from OBDD is in OutputP or +Enum·SR from DT is in OutputP, then enumerating the min- +imal transversals of a hypergraph is in OutputP. +Proof. The proof leans on the proof of Theorem 2 +in [Kavvadias et al., 1993]. Let H be an hypergraph. Vertices +are identified by integers 1, . . . , n and associated to variables +x1, . . . , xn. Let tr(H) be the set of transversals of H and let + +trmin(H) be the set of minimal transversals of H. For each +S ⊆ {1, . . . , n} of vertices let aS be the assignment such that +aS(xi) = 0 if and only if i ∈ S, and let γS = � +i∈S xi. +Observe that aS satisfies γS′ if and only if S ∩ S′ ̸= ∅. Let +f be the function whose satisfying assignments are exactly +the aH for H ∈ H. Denote by sat(f) the set of satisfying +assignments of f. +Now we have the following: +f |= γS ⇔ ∀H ∈ H, aH satisfies γS +⇔ ∀H ∈ H, H ∩ S ̸= ∅ +⇔ S is a transversal of H +This means that the set of implicates of f containing only +negative literals is {γT +| T ∈ tr(H)}, and that the set +of prime implicates of f containing only negative literals is +{γT | T ∈ trmin(H)}. Since the prime implicants of ¬f are +exactly the negation of the prime implicates of f, we get that +the set of prime implicants of ¬f containing only positive lit- +erals is {� +i∈T xi | T ∈ trmin(H)}. Observe that a∅ is the +assignment that set all xi to 1 and that +SR(¬f, a∅) = +�� +i∈T xi | T ∈ trmin(H) +� +. +From H we construct sat(f) in polynomial time. Then from +sat(f) we construct in polynomial time an OBDD circuit B +equivalent to f. Then we obtain an OBDD B′ equivalent to +¬f by switching the 0-sink and the 1-sink of B. Given the +bijection between SR(B′, a∅) and trmin(H), any algorithm for +enumerating sufficient reasons from OBDD can be run with +inputs B′ and a∅ to enumerate the minimal transversals of H. +So if Enum·SR from OBDD is in OutputP then enumerating +the minimal transversals of a hypergraph is in OutputP. +Finally, note that from sat(f) one can construct a decision +tree representing f in polynomial time (instead of an OBDD +circuit), and that negating such a decision tree boils down to +turning 0-leaves into 1-leaves and vice-versa. So the state- +ment also holds for Enum·SR from DT. + diff --git a/XNFQT4oBgHgl3EQfczY1/content/tmp_files/load_file.txt b/XNFQT4oBgHgl3EQfczY1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e5f1258c778fa1416da03974621eb64ed7c4df6c --- /dev/null +++ b/XNFQT4oBgHgl3EQfczY1/content/tmp_files/load_file.txt @@ -0,0 +1,967 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf,len=966 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='13328v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='AI] 30 Jan 2023 On the Complexity of Enumerating Prime Implicants from Decision-DNNF Circuits Alexis de Colnet1 and Pierre Marquis1,2 1Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Artois, CNRS, Centre de Recherche en Informatique de Lens (CRIL), F-62300 Lens, France 2Institut Universitaire de France {decolnet, marquis}@cril.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='fr Abstract We consider the problem Enum·IP of enumerating prime implicants of Boolean functions represented by decision decomposable negation normal form (dec-DNNF) circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We study Enum·IP from dec- DNNF within the framework of enumeration com- plexity and prove that it is in OutputP, the class of output polynomial enumeration problems, and more precisely in IncP, the class of polynomial in- cremental time enumeration problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We then fo- cus on two closely related, but seemingly harder, enumeration problems where further restrictions are put on the prime implicants to be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In the first problem, one is only interested in prime implicants representing subset-minimal abductive explanations, a notion much investigated in AI for more than three decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In the second problem, the target is prime implicants representing suffi- cient reasons, a recent yet important notion in the emerging field of eXplainable AI, since they aim to explain predictions achieved by machine learn- ing classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We provide evidence showing that enumerating specific prime implicants correspond- ing to subset-minimal abductive explanations or to sufficient reasons is not in OutputP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' 1 Introduction Prime implicants are a key concept when dealing with Boolean functions since the notion has been introduced seven decades ago [Quine, 1952].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Within AI, prime implicants (or the dual concept of prime implicates) have been con- sidered for modeling and solving a number of problems, including compiling knowledge [Reiter and de Kleer, 1987] and generating explanations of various kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This is the case in logic-based abductive reasoning (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', [Selman and Levesque, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Eiter and Gottlob, 1995]), a form of inference required in a number of applications when the available knowledge base is incomplete (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', in medicine) and because of such an incompleteness, it cannot alone ex- plain the observations that are made about the state of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Abduction gave rise to much research in AI for the past three decades, especially because it is closely con- nected to other reasoning settings, including truth main- tenance [de Kleer, 1986], assumption-based reasoning and closed-world reasoning (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', [Marquis, 2000] for a sur- vey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Formally, the explanations one looks for are terms over a preset alphabet (composed of the so-called abducible variables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', representing diseases) such that the manifes- tations that are reported (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', some symptoms) are logical consequences of the background knowledge when completed by such a term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In order to avoid trivial explanations, one also asks those terms to be consistent with the knowledge base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Explanations that are the less demanding ones from a logical standpoint (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', subset-minimal ones) can be char- acterized as specific prime implicants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' More recently, de- riving explanations justifying why certain predictions have been made has appeared as essential for ensuring trust- worthy Machine Learning (ML) technologies [Miller, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Molnar, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In the research area of eXplainable AI (XAI), recent work has shown how ML classifiers of various types (including black boxes) can be associated with Boolean circuits (alias transparent or “white” boxes), exhibiting the same input-output behaviours [Narodytska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Shih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Shih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Thanks to such map- pings, XAI queries about classifiers can be delegated to the corresponding circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The notion of sufficient reason of an instance given a Boolean function f modeling a binary classifier has been introduced in [Darwiche and Hirth, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Given an instance a (a simply is an assignment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', a vec- tor of truth values given to each of the n variables) such that f(a) = 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' f(a) = 0), a sufficient reason for a is a subset-minimal partial assignment a′ which is coherent with a (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', a and a′ give the same values to the variables that are assigned in a′) and which satisfies the property that for every extension a′′ of a′ we have f(a′′) = 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' f(a′′) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The features assigned in a′ (and the way they are assigned) can be viewed as explaining why a has been classified by f as a positive (or as a negative) instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Whatever the way prime implicants are used, generating them is in general a computationally demanding task, for at least two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' On the one hand, deriving a single prime implicant of a Boolean function represented by a proposi- tional formula (or circuit) is NP-hard since such a formula is satisfiable when it has a prime implicant, and it is valid precisely when this prime implicant is the empty term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' On the other hand, a source of complexity is the number of prime implicants that may prevent from computing them all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In- deed, it is well-known that the number of prime implicants of a Boolean function can be exponential in the number of variables of the function, and, for many representations of the function, also exponential in the size of the representa- tion (just consider the parity function as a matter of example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In more detail, the number of prime implicants of a Boolean function can be larger than the number of assignments sat- isfying the function [Dunham and Fridshal, 1959];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' there also exist families of Boolean functions over n variables having Ω( 3n n ) prime implicants [Chandra and Markowsky, 1978].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In this paper, we focus on the issue of enumerating prime implicants of a Boolean function represented by a decision decomposable negation normal form circuit (alias a dec- DNNF circuit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The question is to determine whether such prime implicants can be enumerated “efficiently”, which is obviously not the case when the circuit considered is uncon- strained (as explained above, in such a case, computing a sin- gle prime implicant is already hard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This question is impor- tant for all the problems listed previously, when prime impli- cants represent explanations: since they are typically too nu- merous to be computed as a whole, it makes sense to derive them in an incremental way, with some performance guar- antees in the generation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' this lets the user who asked for an explanation deciding what to do after each derivation, namely to stop the enumeration process since he/she is satisfied by the explanation that has been provided, or alternatively to ask for a further explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The dec-DNNF language [Oztok and Darwiche, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Darwiche, 2001] and its subsets FBDD (free binary deci- sion diagrams) [Gergov and Meinel, 1994], OBDD (ordered binary decision diagrams) [Bryant, 1986] and even DT (the set of all binary decision trees over Boolean variables, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', [Wegener, 2000, Chapter 2]) appear at first sight as good candidates for representing the function in the perspec- tive of enumerating “efficiently” its prime implicants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In- deed, they are known as tractable representation languages (they support in polynomial time many queries and trans- formations from the so-called knowledge compilation map [Darwiche and Marquis, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Koriche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 2013]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The main contribution of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We give a polynomial incremental time algorithm for enumer- ating the prime implicants of a Boolean function f repre- sented by a dec-DNNF circuit Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Given Σ and a positive integer k, this algorithm returns k prime implicants of Σ in O(poly(k + |Σ|)) time, or returns all prime implicants of Σ if there are fewer than k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This shows that enumerating prime implicants from dec-DNNF is in the enumeration complex- ity class IncP [Strozecki, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We also provide evidence showing that enumerating specific prime implicants corre- sponding to subset-minimal abductive explanations or to suf- ficient reasons is not in OutputP: on the one hand, comput- ing a single subset-minimal abductive explanation from an OBDD circuit or a decision tree is NP-hard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' on the other hand, the existence of an output polynomial time algorithm for enumerating sufficient reasons from an OBDD circuit or a decision tree would lead to an output polynomial time al- gorithm for enumerating the minimal transversals of a hy- pergraph, thus answering a long-standing question related to monotone dualization [Eiter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 2008].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We start with some preliminaries (Section 2) where the language of dec- DNNF circuits and the framework used to study enumera- tion problems are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We formally define the problem Enum·IP of enumerating prime implicants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then in Section 3 we show that generating the set of all prime implicants from a dec-DNNF circuit is feasible in output polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' From there, we show in Section 4 that Enum·IP from dec- DNNF is in fact in IncP and point out a polynomial incremen- tal time enumeration algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Finally, in Section 5 we focus on subset-minimal abductive explanations and sufficient rea- sons and show that for each of the two cases, the enumeration issue is seemingly harder than in the case when all prime im- plicants are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' All the proofs are reported in a final appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' 2 Preliminaries A Boolean function over n variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , xn is a mapping f from {0, 1}n to {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The set of variables of f is denoted by var(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The assignments to var(f) mapped to 1 by f are called satisfying assignments of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A literal upon variable x is either x or its negation x and a term is a conjunction of literals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We often omit the conjunction symbols when writing terms, for instance we may shorten a ∧ c into a c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We define the empty term t∅ as the term over zero literal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The empty term verifies t∧t∅ = t for every term t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Given ℓ ∈ {x, x}, we denote by f|ℓ the Boolean function over var(f) \\ {x} whose satisfying assignments coincide with that of f ∧ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We use the usual symbols ∧, ∨, ¬, |= to denote conjunction, disjunction, negation, and entailment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Given a set S of terms, max(S, |=) denotes the subset of terms of S that do not entail another term in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' An implicant of a Boolean function f is a term t whose satisfying assignments also satisfy f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', t |= f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' An implicant t is prime when no term t−ℓ obtained by removing a literal ℓ from t is an implicant of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='1 Compilation Languages Compilation languages are often seen as classes of circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let PS be a countable set of propositional variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A circuit in negation normal form (NNF) is a directed acyclic graph (DAG) whose leaves are labelled with 0 (false), 1 (true), or a literal built upon x ∈ PS, and whose internal nodes are labelled with ∧ or ∨ connectives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' we call them ∧- nodes and ∨-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' An NNF circuit computes a Boolean function over the variables appearing in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' For v a node of an NNF circuit Σ, var(v) denotes the set of variables labelling leaves under v in Σ and Σv denotes the subcir- cuit of Σ rooted at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The language of decomposable NNF (DNNF) contains the NNF circuits where ∧-nodes are de- composable, that is, the children v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , vm of every ∧-node v are such that var(vi) ∩ var(vj) = ∅ for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The lan- guage of deterministic, decomposable NNF (d-DNNF) con- tains the DNNF circuits Σ where ∨-nodes are deterministic, that is, the children v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , vm of every ∨-node v are such that Σvi ∧ Σvj is inconsistent for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Finally, the language of decision DNNF (dec-DNNF) is that of circuits whose leaves are labelled with 0, 1, or a literal built upon x ∈ PS, and whose internal nodes are decision nodes and ∧-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Whenever n is a decision node labelled by vari- able x in a dec-DNNF circuit Σ, the circuit Σn given by x u v is viewed as a compact representation of the d- DNNF circuit ∨ ∧ ∧ x x u v (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='6 and Figure 2 in [Darwiche and Marquis, 2002]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Thus, a decision node n is labelled by a variable and has two children: the 0-child (node u on the previous picture) and the 1-child (node v on the previous picture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If n is labelled by x and Σu (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Σv) represents the function f0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' f1), then Σn represents the function (x ∧ f0) ∨ (x ∧ f1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' For instance, Figure 1a gives a dec-DNNF circuit whose deepest decision node computes (s ∧ 1) ∨ (s ∧ p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' It is worth mentioning that all Boolean functions on finitely many variables can be represented in dec-DNNF, or indeed in any of its subsets like FBDD, OBDD, and DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let Σ be a dec-DNNF circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The size of Σ, denoted by |Σ| is its number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' From a dec-DNNF cir- cuit Σ, one can easily derive in polynomial time a dec- DNNF circuit equivalent to Σ where every ∧-node has ex- actly two children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since it is computationally harmless, for the sake of simplicity, our enumeration algorithms suppose that the dec-DNNF circuits satisfy this condition, so that their size is at most twice their number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In the same vein, we suppose that our dec-DNNF circuits have been re- duced, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', every node v in Σ such that Σv computes the 0 function reduces to a leaf labelled by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Testing the sat- isfiability of a dec-DNNF circuit is feasible in linear time [Darwiche and Marquis, 2002], so reducing a dec-DNNF cir- cuit also is a polynomial-time operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='2 Enumeration Complexity We now recall some enumeration complexity classes as de- scribed in [Strozecki, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let V be an alphabet and let A be a binary predicate in V ∗ × V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Given an instance x ∈ V ∗ (the input), A(x) (the set of solutions) denotes the set of all y ∈ V ∗ such that A(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The enumeration problem Enum·A is the function mapping x to A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Enum·A is in the class EnumP if for every y ∈ A(x), |y| is polynomial in |x|, and if deciding whether y is in A(x) is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' EnumP does not capture the complexity of computing the set of solutions A(x), it serves more as a counterpart of NP for enumeration problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The model used for the enumeration of solutions is the ran- dom access machine (RAM) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' See [Strozecki, 2019] for details on why RAM have been chosen for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A RAM solves Enum·A if, for all x, it returns a se- quence y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , ym of pairwise distinct elements such that {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , ym} = A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Enum·A is in OutputP if there is a RAM solving Enum·A in time O(poly(|x| + |A(x)|)) on every input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' OutputP is a relevant enumeration class when the whole set of solutions is explicitly asked for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' For instance, the dualization of a monotone CNF formula φ is the task of generating a DNF formula equivalent to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Because of the monotony condition on φ, the terms used in any smallest DNF formula equivalent to φ are precisely its prime implicants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Thus, the dualization problem boils down to enumerating all the prime implicants of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' For other applications, computing only a fixed number of solutions may be enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A RAM solves Enum·A in in- cremental time f(t)g(n) if on every x, it runs in time time O(f(t)g(|x|)) and returns a sequence y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , yt of t pairwise distinct elements of A(x) when t ≤ |A(x)|, and the whole set A(x) when t > |A(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Enum·A is in IncP if there is a RAM that solves A in incremental time O(tanb) for some constants a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' IncP has a characterization that uses the function problem AnotherSol·A which, given x and S ⊆ A(x), re- turns y ∈ A(x) \\ S when S ̸= A(x), and false otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 1 ([Strozecki, 2019]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A problem Enum·A in EnumP is in IncP if and only if AnotherSol·A is in FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Note that OutputP is thought to be distinct from IncP [Strozecki, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' 3 Enum·IP from dec-DNNF is in OutputP Let us first consider the problem of enumerating the prime implicants of a Boolean function f given as a dec-DNNF cir- cuit Σ, for short the prime implicants of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let IP(Σ, t) be the binary predicate representing the relation that t is a prime im- plicant of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then IP(Σ) denotes the set of prime implicants of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We extend the notation IP(·) to any Boolean function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' To be able to speak of prime implicants enumeration from cir- cuits other than dec-DNNF ones we write “Enum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='IP from L” with L the language Σ belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We start with a couple of easy results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' First of all, since there is a linear-time procedure to verify that a term is an im- plicant of a dec-DNNF circuit, there is a polynomial-time al- gorithm to decide whether a given a term is a prime implicant of a dec-DNNF circuits, thus: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Enum·IP from dec-DNNF is in EnumP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In addition, it is known that Enum·IP from OBDD is in OutputP [Madre and Coudert, 1991], and it is almost straightforward to extend this result to dec-DNNF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' To make it precise, let us briefly describe the output polynomial con- struction of IP(Σ) for Σ, a dec-DNNF circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The construc- tion is based on the three following, folklore propositions (for the sake of completeness, a proof for each of them is nonethe- less reported as a supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let f and g be Boolean functions, then IP(f ∧ g) = max({t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g)}, |=).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Furthermore if var(f) ∩ var(g) = ∅, then IP(f ∧ g) = {t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let f a Boolean function, let x be a variable, and let ℓ ∈ {x, x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Consider t ∈ IP(f|ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If t |= f|ℓ, then t ∈ IP(f), otherwise t ∧ ℓ ∈ IP(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let f be a Boolean function and let x be a variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' IP(f) = {t ∧ x | t ∈ IP(f|x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' t ̸|= f|x} ∪ {t ∧ x | t ∈ IP(f|x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' t ̸|= f|x} ∪ IP(f|x ∧ f|x) h e e ∧ ∧ b p s b s p 1 p (a) A dec-DNNF circuit h v0 : e e v1 : ∧ ∧ b v2 : p s b s v3 : p 1 p S3 = ∅ S0 = {h e b p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' h p s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' e p s} S2 = {b p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' p s} S1 = {e b p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' p s} (b) A path followed in MissingIP h v0 : e e v1 : ∧ ∧ b v2 : p s b s v3 : p 1 p h e b s ∈ IP(Σv0) s ∈ IP(Σv3) e b s ∈ IP(Σv1) b s ∈ IP(Σv2) (c) Propagation of an implicant Figure 1: Generation of a new prime implicant from a dec-DNNF circuit Note that t ∈ IP(f|x) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' IP(f|x)) entails f|x (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' f|x) if and only if t is subsumed by some term in IP(f|x∧f|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' As a consequence, from IP(f|x) and IP(f|x), one can construct IP(f|x∧f|x) in polynomial time thanks to Proposition 3 and we use it to derive IP(f) thanks to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We also have that (see the algorithm for condi- tioning a prime implicant representation provided in [Darwiche and Marquis, 2002]): Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let f a Boolean function and let x be a vari- able, then |IP(f)| ≥ max(|IP(f|x)|, |IP(f|x)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Consider now a dec-DNNF circuit Σ and an internal node v with two children u and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If the sets IP(Σu) and IP(Σw) are provided, then IP(Σv) is obtained in polynomial time us- ing Proposition 3 if v is a decomposable ∧-gate, and using Proposition 5 if v is a decision node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Furthermore, in both cases, we have |IP(Σv)| ≥ max(|IP(Σu)|, |IP(Σw)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' These observations lead to a simple algorithm that generates IP(Σ) by computing the sets IP(Σv) for every node v of Σ consid- ered in a bottom-up way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since constructing the set of prime implicants for any node given that of its children is tractable, since this set is smaller than |IP(Σ)|, and since it is computed only once, the algorithm runs in time O(poly(|Σ|+|IP(Σ)|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Thus, we get: Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Enum·IP from dec-DNNF is in OutputP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We give the construction of the sets of prime implicants for the nodes v1, v2, v3 in the dec-DNNF circuit Σ represented on Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' v3: the sets of prime implicants of the children are IP(1) = {t∅} and IP(p) = {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Using Proposition 5 we have that s ∧ t∅ = s and Σv3|s = p, so s ∧ t∅ ̸|= Σv3|s showing that s ∈ IP(Σv3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We also have that Σv3|s = 1, so s p |= Σv3|s showing that s p ̸∈ IP(Σv3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Finally, we have that IP(Σv3|s ∧ Σv3|s) = {p} by Proposition 3, so IP(Σv3) = {s, p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' v2: the sets of prime implicants of the children are IP(p) = {p} and IP(Σv3) so we compute IP(Σv2) = {b p, b s, b p, p s} v1: the sets of prime implicants of the children are IP(p ∧ s) = {p s} and IP(Σv2) so we compute IP(Σv1) = {e b p, e b p, e b s, p s} 4 Enum·IP from dec-DNNF is in IncP We now investigate Enum·IP from dec-DNNF from the in- cremental enumeration perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Based on Proposition 1, we design a tractable algorithm AnotherIP for solving the problem AnotherSol·IP, thus showing that Enum·IP from dec-DNNF is in IncP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='1 Solving the decision variant of AnotherSol·IP We first consider the decision variant of AnotherSol·IP from dec-DNNF: given a dec-DNNF circuit Σ and a set S ⊆ IP(Σ), return false if and only if S ̸= IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Recall from the discussion preceding Proposition 7 that there is a bottom-up procedure for generating all prime implicants of the dec-DNNF circuit Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' To address the decision variant of AnotherSol·IP on inputs Σ and S, a reverse, top-down search is performed, assuming that S is IP(Σ) until finding a contra- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Before defining what a contradiction means in this setting, a few notations are useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' For t a term and X a set of vari- ables, tX denotes the restriction of t to variables in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Note that if X and var(t) are disjoint, then tX is the empty term t∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let Σ be a dec-DNNF circuit and let S ⊆ IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If the root of Σ is an ∧-node, let u and w be its chil- dren and let Su = {tvar(Σu) | t ∈ S} and Sw = {tvar(Σw) | t ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then Su ⊆ IP(Σu) and Sw ⊆ IP(Σw) hold, and S = IP(Σ) iff Su = IP(Σu) and Sw = IP(Σw) and S = {tu ∧ tw | tu ∈ Su, tw ∈ Sw}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let Σ be a dec-DNNF circuit whose root is a decision node labelled by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let u be its 0-child and w be its 1-child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Given S ⊆ IP(Σ), let Su = {t | t ∧ x ∈ S} ∪ (S ∩ IP(Σu)), Sw = {t | t ∧ x ∈ S} ∪ (S ∩ IP(Σw)) and S′ = {t | t ∈ S, x ̸∈ var(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then Su ⊆ IP(Σu) and Sw ⊆ IP(Σw) hold, and S = IP(Σ) iff Su = IP(Σu) and Sw = IP(Σw) and S′ = max({tu ∧ tw | tu ∈ Su, tw ∈ Sw}, |=).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let v be the root of the dec-DNNF circuit Σ and let S ⊆ IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We say that we have a contradiction at node v when (c1) S = ∅ while Σ is satisfiable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' or (c2) v is a decision node,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Su = IP(Σu) and Sw = IP(Σw),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' but S′ ̸= max({tu ∧ tw | tu ∈ Su,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' tw ∈ Sw},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' |=),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' or Algorithm 1: MissingIP(Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' P) Promises: Σ is reduced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' S ⊆ IP(Σ) 1 Let v be the root of Σ and let P ′ ← P ∪ (v) 2 if λ(v) = |S| then return false 3 if S = ∅ then 4 if v is labelled by 0 then set λ(v) to 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' return false 5 else return (GenerateIP(Σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' P ′) 6 end 7 if v is a ∧-node with children u and w then 8 Build Su and Sw as in Proposition 8 9 r ← MissingIP(Σu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Su,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' P ′) 10 if r ̸= false then return r 11 r ← MissingIP(Σw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Sw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' P ′) 12 if r ̸= false then return r 13 S∗ ← {tu ∧ tv | tu ∈ Su,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' tw ∈ Sw} 14 if S ̸= S∗ then for any t ∈ S∗ \\ S return (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' P ′) 15 else if v is a decision node with children u and w then 16 Build Su,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Sw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' S′ as in Proposition 9 17 r ← MissingIP(Σu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Su,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' P ′) 18 if r ̸= false then return r 19 r ← MissingIP(Σw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Sw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' P ′) 20 if r ̸= false then return r 21 S∗ ← max({tu ∧ tw | tu ∈ Su,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' tw ∈ Sw},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' |=) 22 if S∗ ̸= S′ then for any t ∈ S∗ \\ S′ return (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' P ′) 23 end 24 Set λ(v) to |S| and return false (c3) v is a decomposable ∧-node and Su = IP(Σu) and Sw = IP(Σw) but S ̸= {tu ∧ tw | tu ∈ Su,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' tw ∈ Sw}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A contradiction guarantees that S ̸= IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The contra- diction (c1) is easy to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Contradictions (c2) and (c3) on the other hand require to show that Su = IP(Σu) and Sw = IP(Σw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' When v is an internal node, with children u and w, if there is no contradiction (c1) at v, we use Propo- sitions 8 and 9 to build from S two sets Su and Sw that we recursively compare to IP(Σu) and IP(Σw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Either the re- cursion ends under u or w on a contradiction, in which case S ̸= IP(Σ), or it stops by itself (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', when reaching the leaves of the circuit), which shows that Su = IP(Σu) and Sw = IP(Σw), and then we can check whether there is con- tradiction (c2) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' (c3)) at node v if it is a decision node (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' decomposable node).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If there is none, then S = IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The procedure is given by Algorithm MissingIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The inputs are a dec-DNNF circuit Σ, a set S ⊆ IP(Σ) and a path P in Σ (which will be useful later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A function λ mapping the nodes of Σ to integers is used for memoization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Initially λ(v) = −1 for every node v, but λ(v) may be as- signed a non-negative value at some point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' More precisely, the first time a call MissingIP(Σv, S, P) returns false, we learn that S = IP(Σv) and set λ(v) to |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then for each later call MissingIP(Σv, S′, P ′) with S′ ⊆ IP(Σv), we check whether S′ = IP(Σv) by verifying that λ(v) = |S′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Given a reduced dec-DNNF circuit Σ and S ⊆ IP(Σ), MissingIP(Σ, S, ∅) runs in time O(poly(|S|+ |Σ|)), and it returns false if and only if S = IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Algorithm 2: GenerateIP(Σ) Promise: Σ is satisfiable 1 Find a satisfying assignment a of Σ 2 Let t = � a(x)=1 x ∧ � a(x)=0 x 3 while there is ℓ ∈ t such that t − ℓ |= Σ do 4 Remove ℓ from t 5 end 6 Return t 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='2 Augmenting an incomplete subset of IP(Σ) We build upon MissingIP so that, when S ̸= IP(Σ), we also return a prime implicant in IP(Σ) \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The idea is to use the path P to keep track of the ancestor nodes that were visited before reaching a contradiction and to use P to con- struct a prime implicant in IP(Σ) \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' As an example, con- sider calling MissingIP(Σ, S0, ∅) with Σ the dec-DNNF circuit of Figure 1a and S0 = {h eb p, h p s, e p s} a set of prime implicants of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Figure 1b shows a scenario when MissingIP(Σ, S0, ∅) calls MissingIP(Σv1, S1, (v0)), which calls in turnMissingIP(Σv2, S2, (v0, v1)), which fi- nally calls MissingIP(Σv3, S3, (v0, v1, v2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since S3 = ∅ and Σv3 is reduced and different from 0, the algorithm has reached a contradiction (c1) at node v3 and has not returned false, thus indicating that S0 ̸= IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' MissingIP has fol- lowed the path P = (v0, v1, v2, v3) to reach that contradiction and has kept it in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This path P can then be used to generate a prime implicant in IP(Σ) \\ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' First MissingIP returns the path P to v3 as well as a prime implicant of Σv3, say it is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then we construct a prime implicant of Σv2 upon s, here since v3 is the 0-child of v2 and since s does not entail the 1-child of v2 we obtain b s ∈ IP(Σv2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then we con- struct a prime implicant of Σv1 upon b s, here since since v2 is the 0-child of v1 and since b s does not entail the 1-child of v1 we obtain e b s ∈ IP(Σv1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Repeating the step one more time leads to h e b s ∈ IP(Σv0) = IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The procedure is illustrated in Figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In this example, for generating a new prime implicant of Σ, we have created t ∈ Σv3 \\ S3 and augmented it using Proposition 4 as we travelled backwards along P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We say that we have propagated t along the path P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Accordingly, the algorithm AnotherIP to generate a new prime implicant breaks into two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' First MissingIP(Σ, S, P) searches for a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' It re- turns false if S = IP(Σ) or a pair (t, P) with P the path followed to reach a node v where a contradiction has been found (like v3 in the example), and t a prime of Σv that could not be derived from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The procedure GenerateIP is used to generate t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' GenerateIP runs in polynomial time thanks to linear-time implicant check on dec-DNNF circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Finally PropagateIP is called to propagate t along the path P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The next proposition shows the correctness of AnotherIP: Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let Σ be a reduced dec-DNNF circuit and let S ⊆ IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' AnotherIP(Σ, S) runs in time O(poly(|S| + |Σ|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' It returns false if S = IP(Σ), otherwise it returns a prime implicant of Σ that does not belong to in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' On this basis, the existence of a polynomial incremental Algorithm 3: Propagate(Σ, t, P = (v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , vi)) Promise: Σ is reduced, its root is v0, P is a path in Σ 1 if |P| = 1 then return t 2 if vi−1 is a ∧-node with children u and w then 3 if vi = u then t′ ← GenerateIP(Σw) 4 if vi = w then t′ ← GenerateIP(Σu) 5 else if vi−1 is a decision node for variable x with 0-child u and 1-child w then 6 if vi = u then 7 if t |= Σw then t′ ← t∅ else t′ ← x 8 else 9 if t |= Σu then t′ ← t∅ else t′ ← x 10 end 11 Propagate(Σ, t ∧ t′, (v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , vi−1)) Algorithm 4: AnotherIP(Σ, S) Promise: Σ is reduced, S ⊆ IP(Σ) 1 r ← MissingIP∗(Σ, S, ∅) 2 if r = false then return false 3 else if r = (t, P) then return Propagate(Σ, t, P) time enumeration of prime implicants for dec-DNNF circuits can be easily established: Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Enum·IP from dec-DNNF is in IncP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' 5 Enumerating Specific Prime Implicants For some applications, enumerating all prime implicants of f makes sense, even though there can be exponentially many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We have already mentioned the dualization of monotone CNF formulae as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In this section, we describe two problems that ask for generating only specific prime impli- cants, representing respectively subset-minimal abductive ex- planations and sufficient reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' To illustrate the two notions we use the function f com- puted by the dec-DNNF circuit of Figure 1a as a toy example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' f encodes a very incomplete characterization of human-like creatures in Tolkien’s Middle Earth based on four physical at- tributes: presence of beard and facial hair (b), small size (s), human-like skin (h), pointy ears (p), plus the indication of whether the creature is enrolled in the armies of evil (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We imagine that there are only seven possible creatures: hobbits (h b p s e), elves (h b p s e), dwarfs (h b p s e), men and women (h∗ps ∗),1 ents (h∗p s e), orcs (h b p∗e) and trolls (h b p s e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The satisfying assignments of f describe these creatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Its prime implicants are the smallest combinations of attributes which guarantee the existence of a creature in our Middle Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='1 Abductive Explanations Abductive explanations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', [Selman and Levesque, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Eiter and Gottlob, 1995]) 1∗ denotes that both choices are possible for the variable, typi- cally here humans may fight for evil, humans and ents may or may not have beards, and orcs have a wide range of size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' can be defined as follows: Definition 1 (Abductive explanation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Given a Boolean function f over variables X, a subset H ⊆ X, and a term m on X \\ H, an abductive explanation is a term t on H such that f ∧ t is satisfiable and f ∧ t |= m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The abduction problem asks whether an abductive explana- tion t exists for the input (f, H, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Consider our toy example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We look for combi- nations of physical attributes that guarantee that the creature is evil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This is an abduction problem with H = {h, b, p, s} and m = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' For instance the term h ∧ p is an abductive ex- planation because there exist creatures with pointy ears and a skin that is not human-like, and all of them are evil (in this case only the orcs fit this description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' It is easy to see that an abductive explanation t is in fact an implicant of ¬f ∨m with the conditions that f ∧t is satisfiable and that t is restricted to variables in H (the abducibles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Fur- thermore, since abduction is not a truth-preserving form of in- ference, one is often interested in generating subset-minimal abductive explanations only (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', the logically weakest ab- ductive explanations);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' they correspond to the prime impli- cants of ¬f ∨m such that f ∧t is satisfiable and t is restricted to variables in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Obviously enough, the abduction problem we focus on (the existence of an abductive explanation) is the same, would we consider subset-minimal abductive explanations or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Indeed, deciding whether an abductive explanation exists is equivalent to deciding whether a subset-minimal abductive explanation exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Unfortunately, the condition that only variables in H are allowed in abductive explanations is al- ready too demanding from an enumeration perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Unless P = NP, there is no polynomial-time algorithm which, given an OBDD circuit or a decision tree computing a function f over X and a set Y ⊆ X, decides whether f has an implicant t with var(t) ⊆ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='2 Sufficient Reasons The notion of sufficient reason2 [Darwiche and Hirth, 2020] (aka prime implicant explanation [Shih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 2018b]) is de- fined as follows: Definition 2 (Sufficient reason).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Given a Boolean function f, let a be any assignment to a superset of var(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A suf- ficient reason for a is a prime implicant t of f (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' ¬f) such that a satisfies t, provided that a satisfies f (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' ¬f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The set of all sufficient reasons for a given f is denoted by SR(f, a) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' SR(¬f, a)) when a satisfies f (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' ¬f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Consider again our toy example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' There is no creature which is small, has human-like skin, pointy ears, no facial hair, and is evil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Finding the reasons of why such a creature cannot exist, means finding sufficient reasons for the assignment a defined by a(h) = a(p) = a(s) = a(e) = 1 2This concept is also referred to as “abductive explanations” [Ignatiev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Ignatiev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 2020];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' in the following, we stick to “sufficient reason” to avoid any confusion with the (distinct) concept of abductive explanations as discussed in the previous sec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' and a(b) = 0 given ¬f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In this case h p e ∈ SR(¬f, a) explains why such a creature cannot exist: there are no crea- tures that are evil and have both human-like skin and pointy ears, but there are such creatures that are non-evil (hobbits and elves), and there are evil creatures that have pointy ears (orcs) or human-like skin (men).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' There are other sufficient reasons for a given ¬f, for instance h s e ∈ SR(¬f, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We define the problem Enum·SR similarly to Enum·IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A couple of results about the complexity of computing sufficient reasons have been pointed out for the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Obvi- ously enough, when no assumption is made on the represen- tation of f, computing a single sufficient reason for an assign- ment a is already NP-hard (for pretty much the same reasons as for the prime implicant case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', f is valid iff for any a, the unique sufficient reason for a given f is the empty term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Furthermore, the number of sufficient reasons for an assign- ment a given f can be exponential in the number of variables even when f is represented in DT [Audemard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Contrary to abductive explanations, it is computationally easy to generate a single sufficient reason from SR(Σ, a) when Σ is an OBDD circuit or a decision tree representing f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A greedy algorithm can be used to this end: if a satisfies Σ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' ¬Σ), then start with the canonical term having a as its unique satisfying assignment and remove literals from this term while ensuring that it still is an implicant of Σ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' ¬Σ), until no more literal can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In addi- tion, when Σ is in DT, we can generate in polynomial time a monotone CNF formula Ψ such that IP(Ψ) = SR(Σ, a) (see [Darwiche and Marquis, 2021] for details), and then take advantage of a quasi-polynomial time algorithm for enumer- ating the elements of IP(Ψ) [Gurvich and Khachiyan, 1999].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Contrastingly, deciding whether a preset number of sufficient reasons for a given a exists is intractable (NP-hard), even when the Boolean function f is monotone (see Theorem 3 in [Marques-Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 2021]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In the following, we complete those results by provid- ing evidence that Enum·SR from any language among dec- DNNF, OBDD, or DT is a difficult problem, despite the fact that those languages are quite convenient for many reasoning tasks [Darwiche and Marquis, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Koriche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let us first give an inductive computation of SR(Σ, a) sim- ilar to that of IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let f and g be Boolean functions with var(f) ∩ var(g) = ∅ and let a be a truth assignment to a superset of var(f) ∪ var(g), then SR(f ∧ g, a) = {t ∧ t′ | t ∈ SR(f, a), t′ ∈ SR(g, a)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let f be a Boolean function, let a be a truth assignment to a superset of var(f) and let x ∈ var(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If a satisfies the literal ℓ on variable x then SR(f, a) = {t ∧ ℓ | t ∈ SR(f|ℓ, a), t ̸|= f|ℓ} ∪ SR(f|x ∧ f|x, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' By Proposition 6, |IP(f)| ≥ max(|IP(f|x)|, |IP(f|x)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In a sense this means that using IP(f|x) and IP(f|x) to generate IP(f) is not a waste of resources since all these implicants are kept in some form through IP(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This led to our out- put polynomial procedure to generate IP(f) for OBDD and more generally for dec-DNNF circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' On the other hand, it is not guaranteed that SR(f, a) is larger than SR(f|x, a) and SR(f|x, a) so there is no straightforward adaptation of this procedure from Enum·IP to Enum·SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let Σ be the dec-DNNF circuit of Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Consider the dec-DNNF circuit Σv1 rooted at node v1, as spotted in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The assignment a to {b, e, p, s} de- fined by a(b) = a(e) = 1 and a(p) = a(s) = 0 satisfies Σv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Recall that the set IP(Σv1) has been con- structed in Example 1 and observe that SR(Σv1, a) = {p s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now the 0-child of v1 is v2 and looking at the set IP(Σv2) constructed in Example 1, we see that SR(Σv2, a) = {p s, b p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since Σv2 = Σv1|e, we have that |SR(Σv1, a)| < max(|SR(Σv1|e, a)|, |SR(Σv1|e, a)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Actually, we give evidence that enumerating sufficient rea- sons from dec-DNNF, and even from OBDD or DT, is not in OutputP by reducing to it the problem of enumerating the minimal transversals of a hypergraph, a well-known problem whose membership to OutputP is a long-standing question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Formally: Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If Enum·SR from OBDD is in OutputP or Enum·SR from DT is in OutputP, then enumerating the min- imal transversals of a hypergraph is in OutputP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' 6 Conclusion Most applications of prime implicants for Boolean function analysis use only a fraction of the many prime implicants a Boolean function may have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Especially, in the context of logic-based abduction, subset-minimal assumptions to be added to the available background knowledge in order to be able to derive some given manifestations are looked for;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' in the propositional case, they correspond to specific prime im- plicants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Furthermore, in an eXplainable AI perspective, spe- cific prime implicants known as sufficient reasons are used to explain the predictions of machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In our work, we have studied the enumeration of general and specific prime implicants of Boolean functions repre- sented as dec-DNNF circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' It was known that these circuits enable efficient reasoning on Boolean functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Our results show that when it comes to prime implicants enumeration, dec-DNNF circuits have benefits as well as limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Our take-home message is that, while dec-DNNF circuits enable enumerating general prime implicants in incremental polyno- mial time, there are strong pieces of evidence against the exis- tence of any output-polynomial time procedure for enumerat- ing specific prime implicants from dec-DNNF circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' More precisely, if a procedure for enumerating subset-minimal ab- ductive explanations were to exist, then P = NP would fol- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Similarly, if there were an output-polynomial time al- gorithm for enumerating sufficient reasons from dec-DNNF circuits, then the enumeration of the minimal transversals of a hypergraphwould be in OutputP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Though this is considered unlikely in enumeration complexity, we think that proving a stronger statement would be a valuable contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We let this task open for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Acknowledgments Many thanks to the anonymous reviewers for their comments and insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This work has benefited from the supports of the PING/ACK project (ANR-18-CE40-0011) and of the AI Chair EXPEKCTATION (ANR-19-CHIA-0005-01)of the French National Research Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' It was also partially sup- ported by TAILOR, a 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' of AAAI’90, pages 343–348, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' [Shih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 2018a] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Shih, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Choi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Darwiche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' For- mal verification of Bayesian network classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' of PGM’18, pages 427–438, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' [Shih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 2018b] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Shih, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Choi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Darwiche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A symbolic approach to explaining Bayesian network classi- fiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' of IJCAI’18, pages 5103–5111, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' [Shih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 2019] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Shih, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Choi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Darwiche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Com- piling Bayesian networks into decision graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' of AAAI’19, pages 7966–7974, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' [Strozecki, 2019] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Strozecki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Enumeration complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' EATCS, 129, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' [Wegener, 2000] Ingo Wegener.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Branching Programs and Binary Decision Diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' SIAM, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Appendix: Proofs Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Enum·IP from dec-DNNF is in EnumP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Direct from the fact that dec-DNNF is a sublanguage of deterministic DNNF (d-DNNF) and d-DNNF supports polynomial time implicant check [Darwiche and Marquis, 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let f and g be Boolean functions, then IP(f ∧ g) = max({t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g)}, |=).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Furthermore if var(f) ∩ var(g) = ∅, then IP(f ∧ g) = {t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A proof of IP(f ∧ g) = max({t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g)}, |=) can be found e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', in [Marquis, 1993].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In the case where var(f) ∩ var(g) = ∅, the terms in IP(f) contain only variables from var(f) and the terms in IP(g) contain only variables from var(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We denote lit(f) = {x, x | x ∈ var(f)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let tf, t′ f ∈ IP(f) and tg, t′ g ∈ IP(g) such that tf ∧ tg |= t′ f ∧ t′ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Looking at terms as sets of liter- als this means that t′ f ∪ t′ g ⊆ tf ∪ tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' But then t′ f ⊆ tf since (tf ∪tg)∩lit(f) = tf and (t′ f ∪t′ g)∩lit(f) = t′ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This means that tf |= t′ f and therefore tf = t′ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A similar argument gives that tg = t′ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This shows that when var(f) ∩ var(g) = ∅, IP(f ∧ g) = max({t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g)}, |=) = {t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let f a Boolean function, let x be a variable, and let ℓ ∈ {x, x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Consider t ∈ IP(f|ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If t |= f|ℓ then t ∈ IP(f), otherwise t ∧ ℓ ∈ IP(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Suppose t |= f|ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then t is an implicant of f since t |= (f|x∧f|x) |= ((x∧f|x)∨(x∧f|x)) ≡ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' To prove that it is prime let t′ be a strict subterm of t and assume t′ |= f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We have x ̸∈ var(t) since x ̸∈ var(f|ℓ), so t′|ℓ = t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If t′ |= f then t′ = t′|ℓ |= f|ℓ and t is not a prime implicant of f|ℓ, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Suppose t ̸|= f|ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then t ∧ ℓ is an implicant of f since t ∧ ℓ |= (ℓ ∧ f|ℓ) |= ((x ∧ f|x) ∨ (x ∧ f|x)) ≡ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' To prove that it is prime, let t′ be a strict subterm of t ∧ ℓ and assume t′ |= f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If ℓ ̸∈ t′ then t′ = t′|ℓ |= f|ℓ and t is not a prime implicant of f|ℓ, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If however ℓ ∈ t′, then write t′ = t′′ ∧ ℓ and observe that t′′ = t′|ℓ |= f|ℓ, so t is not a prime implicant of f|ℓ, another contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let f be a Boolean function and let x be a variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' IP(f) = {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} ∪ {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} ∪ IP(f|x ∧ f|x) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We derive {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} ∪ {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} ⊆ IP(f) from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now we show that IP(f|x ∧ f|x) ⊆ IP(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let t ∈ IP(f|x ∧ f|x), then t |= f|x and t |= f|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since x ̸∈ var(f|x)∪var(f|x) we have that x ̸∈ var(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' First we prove that t is an implicant of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If we had t ̸|= f then t|x ̸|= f|x or t|x ̸|= f|x would hold, but t|x = t|x = t so t |= f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now to prove that it is a prime implicant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let ℓ ∈ t and let t′ be the term t deprived from ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If t′ |= f were to hold then so would t′|x |= f|x and t′|x |= f|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' But since t′|x = t′|x = t′, this would mean that t′ |= f|x ∧ f|x and therefore t would not be a prime implicant of f|x ∧ f|x, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This shows that IP(f|x ∧ f|x) ⊆ IP(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We have established that IP(f) ⊇ {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} ∪ {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} ∪ IP(f|x ∧ f|x) and now we show the reverse inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let t ∈ IP(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' As- sume t = t0 ∧ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' t |= f implies that t|x |= f|x, or in other words, that t0 |= f|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If t0 was not a prime implicant of f|x, that is, if there was t′ 0 ⊂ t0 such that t′ 0 |= f|x, then we would also have t′ 0 ∧ x |= f and therefore t would not be a prime implicant of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So t0 ∈ IP(f|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now if t0 |= f|x then we would have t0 |= f|x ∧ f|x |= f, so t would not be a prime implicant of f, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This shows that if x is in t, then t ∈ {t0 ∧ x | t0 ∈ IP(f|x), t0 ̸|= f|x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A symmetrical proof gives that if x is in t, then t ∈ {t1 ∧ x | t1 ∈ IP(f|x), t1 ̸|= f|x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Finally if neither x nor x is in t, then t = t|x |= f|x and t = t|x |= f|x, and therefore t |= f|x∧f|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now if t was not a prime implicant of f|x∧f|x then there would be some t′ ⊂ t in IP(f|x ∧ f|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since IP(f|x ∧ f|x) ⊆ IP(f), this would mean that t is not a prime implicant of f, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So t ∈ IP(f|x ∧ f|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We have established that IP(f) ⊆ {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} ∪ {t ∧ x | t ∈ IP(f|x), t ̸|= f|x} ∪ IP(f|x ∧ f|x) thus finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let f a Boolean function and let x be a vari- able, then |IP(f)| ≥ max(|IP(f|x)|, |IP(f|x)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let ℓ ∈ {x, x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' It is shown in [Darwiche and Marquis, 2002] that IP(f|ℓ) = max({t|ℓ | t ∈ IP(f)}, |=).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So |IP(f|ℓ)| = | max({t|ℓ | t ∈ IP(f)}, |= )| ≤ |{t|ℓ | t ∈ IP(f)}| = |IP(f)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Enum·IP from dec-DNNF is in OutputP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Given a dec-DNNF circuit Σ, we construct IP(Σ) by visiting every node v of Σ in a bottom-up order while com- puting IP(Σv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We start from the leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If v is labelled by a literal ℓ then IP(Σv) = {ℓ}, if it is labelled by 0 then IP(Σv) = ∅, and if it is labelled by 1 then IP(Σv) = {t∅} where t∅ is the term containing no literal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now let v be an internal node of Σ and let u and w be its children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since we visit the nodes in depth-first order, IP(Σu) and IP(Σw) have already been computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If v is a decomposable ∧-node then using Proposition 3 we com- pute IP(Σv) = {tu ∧ tw | tu ∈ IP(Σu), tw ∈ IP(Σw)} in time O(|IP(Σu)| × |IP(Σw)|) = O(|IP(Σ)|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Observe that |IP(Σv)| ≥ max(|IP(Σu)|, |IP(Σw)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If v is a decision node for the variable x whose 0- and 1- children are u and w, respectively, then we compute IP(Σv) from IP(Σu) and IP(Σw) using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since dec- DNNFs support linear-time implicant check, {t ∧ x | t ∈ IP(Σu), t ̸|= Σw} and {t ∧ x | t ∈ IP(Σw), t ̸|= Σu} are build in time polynomial in |Σ| + |IP(Σu)| + |IP(Σw)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' As for IP(Σu∧Σw), we build it in time polynomial in |IP(Σu)|+ |IP(Σw)| using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Observe that, by Proposition 6, there is again |IP(Σv)| ≥ max(|IP(Σu)|, |IP(Σw)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' When we reach the root node r we compute IP(Σr) = IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since for every node v with children u and w we have that |IP(Σv)| ≥ max(|IP(Σu)|, |IP(Σw)|), we also have that |IP(Σv)| ≤ |IP(Σ)| for every v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We build IP(Σv) only once and the time spent on each node v to build IP(Σv) given IP(Σu) and IP(Σw) is polynomial in |Σ|+|IP(Σ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Summing over all nodes we get that the time needed to build IP(Σ) is also polynomial in |Σ| + |IP(Σ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let Σ be a dec-DNNF circuit and let S ⊆ IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If the root of Σ is an ∧-node, let u and w be its chil- dren and let Su := {tvar(Σu) | t ∈ S} and Sw := {tvar(Σw) | t ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then Su ⊆ IP(Σu) and Sw ⊆ IP(Σw) hold, and S = IP(Σ) iff Su = IP(Σu) and Sw = IP(Σw) and S = {tu ∧ tv | tu ∈ Su, tv ∈ Sv}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If S = IP(Σ) then by Proposition 3 S = {tu ∧ tv | tu ∈ IP(Σu), tv ∈ IP(Σv)} so IP(Σu) = {tvar(Σu) | t ∈ S} = Su and IP(Σv) = {tvar(Σv) | t ∈ S} = Sv and thus S = {tu ∧ tv | tu ∈ Su, tv ∈ Sv}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If S ̸= IP(Σ), let t∗ ∈ IP(Σ) \\ S and let t∗ u = t∗ var(Σu) and t∗ v = t∗ var(Σv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' By Proposition 3, t∗ u (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' t∗ v) is in IP(Σu) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' IP(Σv)), so either t∗ u ̸∈ Su or t∗ v ̸∈ Svand we are done, or t∗ u ∈ Su and t∗ v ∈ Sv in which case {tu∧tv | tu ∈ Su, tv ∈ Sv} ̸= S since t∗ u ∧ t∗ v is in {tu ∧ tv | tu ∈ Su, tv ∈ Sv} but not in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let Σ be a dec-DNNF circuit whose root is a decision node labelled by x and whose 0- and 1-child are u and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Given S ⊆ IP(Σ), let Su = {t | t ∧ x ∈ S} ∪ (S ∩ IP(Σu)), Sw = {t | t ∧ x ∈ S} ∪ (S ∩ IP(Σw)), S′ = {t | t ∈ S, var(x) ̸∈ var(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then Su ⊆ IP(Σu) and Sw ⊆ IP(Σw) hold, and S = IP(Σ) iff Su = IP(Σu) and Sw = IP(Σw) and S′ = max({tu ∧ tw | tu ∈ Su, tw ∈ Sw}, |=).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Algorithm 5: GenerateIP(Σ) Promise: Σ is satisfiable 1 Find a satisfying assignment a of Σ 2 Let t be the corresponding term: t = � a(x)=1 x ∧ � a(x)=0 x 3 while there is ℓ ∈ t such that t − ℓ |= Σ do 4 Remove ℓ from t 5 end 6 return t Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' For convenience we denote S∗ = max({tu ∧ tw | tu ∈ Su, tw ∈ Sw}, |=).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' First we prove that Sw ⊆ IP(Σw) (the proof that Su ⊆ IP(Σu) is analogous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Clearly S ∩ IP(Σw) ⊆ IP(Σw) so we just need to show that {t | t ∧ x ∈ S} ⊆ IP(Σw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let t ∧ x be in S, then t ∧ x |= Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' t is an implicant of Σw since t ≡ (t ∧ x)|x |= Σ|x ≡ Σw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now if there exists t′ ̸= t such that t |= t′ |= Σw then t ∧ x |= t′ ∧ x |= Σ holds, and therefore t ∧ x is not a prime implicant of Σ, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So {t | t ∧ x ∈ S} ⊆ IP(Σw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Second we prove that Sw ̸= IP(Σw) implies S ̸= IP(Σ) (the proof is similar for Su ̸= IP(Σu)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Assume there ex- ists t ∈ IP(Σw) \\ Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If t |= Σu then t is in IP(Σ) by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' But t cannot be in S for otherwise it would be in S ∩ IP(Σw) ⊆ Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This shows that S ̸= IP(Σ) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If however t ̸|= Σu then t ∧ x is in IP(Σ) by Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' But t ∧ x cannot be in S for otherwise t would be in {τ | τ ∧ x ∈ S} ⊆ Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So again S ̸= IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now we prove that (S′ ̸= S∗) ⇒ (S ̸= IP(Σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We may assume that Su = IP(Σu) and Sw = IP(Σw), otherwise S ̸= IP(Σ) holds regardless of S′ = S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since Σu ≡ Σ|x and Σw = Σ|x we have that S∗ = max({tu ∧ tw | tu ∈ IP(Σ|x), tw ∈ IP(Σ|x)}, |=) = IP(Σ|x ∧ Σ|x) by Proposi- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now S = S′ ∪ {t | t ∈ S, x ∈ t} ∪ {t | t ∈ S, x ∈ t} so, by Proposition 5, if S = IP(Σ) then S′ corresponds to the set IP(Σ|x ∧ Σ|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So (S′ ̸= S∗) ⇒ (S′ ̸= IP(Σ|x ∧ Σ|x)) ⇒ (S ̸= IP(Σ)) Now for the other direction, assume there exists t ∈ IP(Σ) \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' First suppose that t = t′ ∧ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' On the one hand t′ is in IP(Σ|x) = IP(Σw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' On the other hand t′ is clearly not in {τ | τ ∧ x ∈ S}, and since it is not an implicant of Σ, it is not in S ∩ IP(Σw) either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This means that t′ ∈ IP(Σw) \\ Sw and therefore Sw ̸= IP(Σw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In the case where t = t′ ∧ x, a similar proof gives that Su ̸= IP(Σu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' It remains to consider the situation where neither x nor x is in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' By Proposition 5, t is contained in IP(Σ|x∧Σ|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' As before, we can assume that Su = IP(Σu) and Sw = IP(Σw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We have already explained that this assumption yields S∗ = IP(Σ|x ∧ Σ|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since t is not in S and x ̸∈ t and x ̸∈ t, we have that t ̸∈ S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So t ∈ S∗ \\ S′, and therefore S ̸= S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Given a reduced dec-DNNF circuit Σ and S ⊆ IP(Σ), MissingIP(Σ, S, ∅) runs in time O(poly(|S|+ |Σ|)), and it returns false if and only if S = IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Soundness: We prove soundness by induction on the depth of Σ using Propositions 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If Σ has depth 1 then it is a single node v labelled by 0, 1 or a literal ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The promise states that S ⊆ IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If v is labelled by 0, then S must be ∅ and the algorithm returns false at line 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If v is is labelled by 1 then either S = {t∅} = IP(1) and the algoritm returns false at line 24, or S = ∅ and the algorithm returns (1, (v)) at line 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Finally if v is labelled by ℓ, either S = {ℓ} = IP(ℓ) and the algorithm returns false at line 24, or S = ∅ and the algorithm returns (ℓ, (v)) at line 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' In all cases the algorithm returns false if and only if S = IP(Σ), and it sets λ(v) to |IP(Σv)| before returning false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now if Σ has depth more than 1, its root node v is ei- ther a decomposable ∧-node or a decision node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since Σ is reduced, it cannot be unsatisfiable, so if S = ∅ the al- gorithm returns (t, (v)) with t ∈ IP(Σ) at line 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' From now on we suppose that S ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If v is a decomposable ∧-node with children u and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' By Proposition 8, since we are promised that S ⊆ IP(Σ), we have that IP(Σ) = S if and only if IP(Σu) = Su and IP(Σw) = Sw and we can construct S from Su and Sw as shown in Proposition 8 (Su and Sw defined as in Proposition 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' By induction IP(Σu) ̸= Su or IP(Σw) ̸= Sw if and only if the output of MissingIP(Σu, Su, ∗) or MissingIP(Σw, Sw, ∗) is dis- tinct from false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So if IP(Σu) ̸= Su or IP(Σw) ̸= Sw, a return statement occurs line 9 or 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Otherwise, it possible that IP(Σu) = Su or IP(Σw) = Sw but that S can not be constructed from Su and Sw, then the return statement of line 14 is triggerd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So if S ̸= IP(Σ), then lines 8-14 return some- thing that is not false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' And if S = IP(Σ), then no return call is triggered lines 8-14 and the algorithm returns false at line 24 after setting λ(v) to |S| = |IP(Σ)| = |IP(Σv)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If v is a decision node for variable x with 0-child u and 1-child w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' By Proposition 9, since we are promised that S ⊆ IP(Σ), we have that IP(Σ) = S if and only if IP(Σu) = Su and IP(Σw) = Sw and S′ = S∗ with Su, Sw and S′ de- fined as in Proposition 9 and S∗ defined line 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' By induction IP(Σu) = Su and IP(Σw) = Sw if and only if the output of MissingIP(Σu, Su, ∗) or MissingIP(Σw, Sw, ∗) is dis- tinct from false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So if S ̸= IP(Σ), then lines 16-22 return something that is not false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' And if S = IP(Σ), then no return call is triggered lines 16-22 and the algorithm returns false at line 24 after setting λ(v) to |S| = |IP(Σ)| = |IP(Σv)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Running time: Consider the time spent in MissingIP(Σ, S, P) before a return statement or a recursive call is triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The procedure may end at line 2 or 4 in O(1) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' It can also end line 5, in which case it has to compute a prime implicant of Σ using GenerateIP, which runs in time O(poly(|Σ|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now if the algorithm has not returned lines 2, 4 or 5, most of the running time is spent building sets of terms from S lines 8, 13, 16 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Building Su and Sw line 8 only requires projecting the terms in S onto var(Σu) and var(Σw), which takes time O(|S|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Construct- ing the set S∗ at line 13 takes O(|Su| × |Sw|) = O(|S|2) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' At line 16, S′ can clearly be obtained in time O(|S|) and Su and Sw are obtained in time O(poly(|S| + |Σ|)) thanks to polynomial-time prime implicant check on dec- DNNF circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Finally the set S∗ at line 21 is constructed in O(|Su| × |Sw|) = O(|S|2) and compared to S′ in time O(poly(|S|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So before a return statement or a recursive call is triggered, the algorithm spends O(poly(|S| + |Σ|) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' One can observe that |Su|, |Sw| are fewer than |S|, so for every node v in Σ, a call MissingIP(Σv, S′, ∗) takes O(poly(|S| + |Σ|)) time before triggering a return statement or a recursive call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Thanks to memoization – implemented via λ – the O(poly(|S| + |Σ|)) time procedure is done only once per node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So the total running time of the algorithm is also in O(poly(|S| + |Σ|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let Σ be a reduced dec-DNNF circuit and let S ⊆ IP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' AnotherIP(Σ, S) runs in time O(poly(|S| + |Σ|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' It returns false if S = IP(Σ), otherwise it returns a prime implicant of Σ that does not belong to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Soundness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' First AnotherIP(Σ, S) calls MissingIP(Σ, S, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Soundness of MissingIP has been established in Proposition 10 so if S = IP(Σ) then MissingIP(Σ, S, ∅) returns false and so does AnotherIP(Σ, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now let us assume that MissingIP(Σ, S, ∅) has not returned false but the pair (t, P) with P = (v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , vm) a path from v0 (the root of Σ) to vm and t a term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Use the notation Pi = (v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , vi−1) for all 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then calling MissingIP(Σ, S, ∅) has triggered a se- quence of recursive calls MissingIP(Σv1, S1, P1), MissingIP(Σv2, S2, P2),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' ,MissingIP(Σvm, Sm, Pm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' A contradiction has been found during the last step: MissingIP(Σvm, Sm, Pm) ended line 5 for a contradiction of type (c1), or line 20 for a contradiction of type (c2), and returned (t, P) with t some term that we claim is in IP(Σvm) \\ Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' t ∈ IP(Σvm) \\ Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This is clear if MissingIP(Σvm, Sm, Pm) ends line 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now if it ends line 20, then vm is a decision node for x with 0-child u and 1-child w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The sets Su, Sw, S′ and S∗ have been generated and that it has been shown that Su = IP(Σu) and Sw = IP(Σw) (otherwise a return statement line 16 or 18 would have been triggered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So S∗ = IP(Σu ∧ Σw) = IP(Σvm|x∧Σvm|x) by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We have t ∈ S∗ \\S′ so it is clear that x ̸∈ var(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Furthermore S′ contains all terms from Sm in which neither x nor x appears, so t ∈ S∗ \\ S′ really means that t ∈ S∗ \\Sm = IP(Σvm|x∧Σvm|x)\\Sm ⊆ IP(Σvm) \\ Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now AnotherIP(Σ, S) returns the result Propagate(Σ, t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' To prove that the output is a term in IP(Σ) \\ S, it is sufficient to show that, for every 1 ≤ i ≤ m, if ti ∈ IP(Σvi) \\ Si then Propagate(Σ, ti, Pi) calls Propagate(Σ, ti−1, Pi−1) with ti−1 ∈ IP(Σvi−1) \\ Si−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The rest is an easy induction (with S0 = S and Σv0 = Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let ti ∈ IP(Σvi) \\ Si with i ≥ 1 then Propagate(Σ, ti, Pi) does a recursive call Propagate(Σ, ti−1, Pi−1) with ti−1 ∈ IP(Σvi−1) \\ Si−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Propagate(Σ, ti, Pi) calls Propagate(Σ, ti ∧ t′, Pi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let ti−1 = ti ∧ t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' We need to show that it is in IP(Σvi−1) \\ Si−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' First assume that vi−1 is a de- composable ∧-node with children vi and w, then t′ is ob- tained line 4 and clearly t′ ∈ IP(Σw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' By Proposition 3, ti ∧ t′ ∈ IP(Σvi ∧ Σw) = IP(Σvi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' By construction Si = {tvar(Σvi) | t ∈ Si−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If ti ∧ t′ was in Si−1 then its restriction ti to var(Σvi) would be Si, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So ti ∧ t′ ̸∈ Si−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now suppose vi−1 is a decision node for x with 0-child u and 1-child w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let vi be u (the case vi = w is analogous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' By construction Si = Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' t′ is obtained line 7 and, by Propo- sition 4, ti ∧ t′ ∈ IP(Σvi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' To prove that ti ∧ t′ ̸∈ Si−1, first assume that ti |= Σw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then t′ is the empty term t∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So ti ∧ t′ = ti and ti ∈ IP(Σvi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If ti was in Si−1 then we would have ti ∈ Si−1 ∩ IP(Σvi) ⊆ Si, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So when ti |= Σw, we have ti ∧ t′ ∈ IP(Σvi−1) \\ Si−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now if ti ̸|= Σw, then ti ∧ t′ = ti ∧ x and ti ∧ x is not in Si−1 for otherwise we would have ti ∈ {τ | τ ∧ x ∈ Si−1} ⊆ Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So again we have ti ∧ t′ ∈ IP(Σvi−1) \\ Si−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' It has already been proved in Proposition 10 that Missing(Σ,S,∅) runs in time O(poly(|S| + |Σ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' As for Propagate(Σ, t, P), |P| recursive calls are made and the cost between two consecutive recursive calls is either one call to GenerateIP line 3 or 4, or one implicant check line 7 or 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' An implicant test on a dec-DNNF takes linear time and GenerateIP makes at most |var(Σ)| such tests, so it runs in time O(poly(|Σ|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Thus Propagate(Σ, t, P) runs in time O(|P| × poly(|Σ|)) = O(poly(|Σ|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Enum·IP from dec-DNNF is in IncP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Using Proposition 11, k prime implicants of Σ can be generated in time O(poly(k + |Σ|)) by simply calling AnotherIP(Σ, S) k times, each time adding to S the new prime implicant that has been computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' This shows that Enum·IP from dec-DNNF is in IncP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Unless P = NP, there is no polynomial-time algorithm which, given an OBDD circuit or a decision tree computing a function f over X and a set Y ⊆ X, decides whether f has an implicant t with var(t) ⊆ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let φ be a CNF formula with m clauses c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Create m fresh variables z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , Bm be OBDD circuits respecting the same variable ordering and computing c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' These OBDD circuits can be computed in polynomial time (and can even be chosen in DT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Define now the OBDD circuits B(i) = (zi ∧ Bi) ∨ (zi ∧ B(i+1)) for 1 ≤ i ≤ m, with B(m+1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' B(1) is an OBDD circuit on {z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , zm} ∪ var(φ) built in polynomial time from φ and whose size is in O(|φ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' An implicant t of B(1) such that var(t) ⊆ var(φ) exists if and only if φ is satisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' For the first direction assume the implicant exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' t is an implicant of B(1) = (z1 ∧ B1) ∨ (z1 ∧ B(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since z1 ̸∈ var(t), we have t |= B1 ≡ c1 and t |= B(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Following the same line of reasoning with B(2) instead of B(1) we also have that t |= B2 ≡ c2 and t |= B(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' And we repeat the argument until reaching, t |= c1, t |= c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , t |= cm, t |= B(m+1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So indeed t |= φ and then φ is satisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' For the other direction assume φ is satisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then there exists an implicant t of φ with var(t) ⊆ var(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let a be a truth assignment to var(φ) ∪ {z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , zm} that satisfies t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If a(zi) = 1 for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', m}, then B(1)|a ≡ B(2)|a ≡ · · ≡ B(m+1)|a ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Otherwise let j be the smallest inte- ger such that a(zj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then B(1)|a ≡ B(2)|a ≡ · · · ≡ B(j)|a ≡ Bj|a ≡ cj|a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since t is an implicant of φ, we have that t |= cj, so cj|a ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Thus every assignment a that sat- isfies t also satisfies B(1), in other words t is an implicant of B(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So if the algorithm from the proposition statement exists, we can run it on inputs B(1) and Y = var(φ) to decide in polynomial time whether φ is satisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Finally note that if one had chosen to represent B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , Bm as decision trees from DT (which is also fea- sible in polynomial time), then B(1) would be an element of DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So the statement also holds for DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let f and g be Boolean functions with var(f) ∩ var(g) = ∅ and let a be a truth assignment to a superset of var(f) ∪ var(g), then SR(f ∧ g, a) = {t ∧ t′ | t ∈ SR(f, a), t′ ∈ SR(g, a)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Comes from Proposition 3: SR(f ∧ g, a) = {τ ∈ IP(f ∧ g) | a satisfies τ} = {t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g), a satisfies t ∧ t′} = {t ∧ t′ | t ∈ IP(f), t′ ∈ IP(g), a satisfies both t and t′} = {t ∧ t′ | t ∈ SR(f, a), t′ ∈ SR(g, a)} Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let f be a Boolean function, let a be a truth assignment to a superset of var(f) and let x ∈ var(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If a satisfies the literal ℓ on variable x then SR(f, a) = {t ∧ ℓ | t ∈ SR(f|ℓ, a), t ̸|= f|ℓ} ∪ SR(f|x ∧ f|x, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Comes from Proposition 5: SR(f, a) ={t ∈ IP(f) | a satisfies t} ={t ∧ ℓ | t ∈ IP(f|ℓ), t ̸|= f|ℓ, a satisfies t} ∪ {t ∧ ℓ | t ∈ IP(f|ℓ), t ̸|= f|ℓ, a satisfies t} ∪ {t ∈ IP(f|x ∧ f|x) |, a satisfies t} ={t ∧ ℓ | t ∈ SR(f|ℓ, a), t ̸|= f|ℓ} ∪ SR(f|x ∧ f|x, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' If Enum·SR from OBDD is in OutputP or Enum·SR from DT is in OutputP, then enumerating the min- imal transversals of a hypergraph is in OutputP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' The proof leans on the proof of Theorem 2 in [Kavvadias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=', 1993].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let H be an hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Vertices are identified by integers 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , n and associated to variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let tr(H) be the set of transversals of H and let trmin(H) be the set of minimal transversals of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' For each S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' , n} of vertices let aS be the assignment such that aS(xi) = 0 if and only if i ∈ S, and let γS = � i∈S xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Observe that aS satisfies γS′ if and only if S ∩ S′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Let f be the function whose satisfying assignments are exactly the aH for H ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Denote by sat(f) the set of satisfying assignments of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Now we have the following: f |= γS ⇔ ∀H ∈ H, aH satisfies γS ⇔ ∀H ∈ H, H ∩ S ̸= ∅ ⇔ S is a transversal of H This means that the set of implicates of f containing only negative literals is {γT | T ∈ tr(H)}, and that the set of prime implicates of f containing only negative literals is {γT | T ∈ trmin(H)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Since the prime implicants of ¬f are exactly the negation of the prime implicates of f, we get that the set of prime implicants of ¬f containing only positive lit- erals is {� i∈T xi | T ∈ trmin(H)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Observe that a∅ is the assignment that set all xi to 1 and that SR(¬f, a∅) = �� i∈T xi | T ∈ trmin(H) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' From H we construct sat(f) in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then from sat(f) we construct in polynomial time an OBDD circuit B equivalent to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Then we obtain an OBDD B′ equivalent to ¬f by switching the 0-sink and the 1-sink of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Given the bijection between SR(B′, a∅) and trmin(H), any algorithm for enumerating sufficient reasons from OBDD can be run with inputs B′ and a∅ to enumerate the minimal transversals of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So if Enum·SR from OBDD is in OutputP then enumerating the minimal transversals of a hypergraph is in OutputP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' Finally, note that from sat(f) one can construct a decision tree representing f in polynomial time (instead of an OBDD circuit), and that negating such a decision tree boils down to turning 0-leaves into 1-leaves and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} +page_content=' So the state- ment also holds for Enum·SR from DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFQT4oBgHgl3EQfczY1/content/2301.13328v1.pdf'} diff --git a/XtE3T4oBgHgl3EQfFwnp/content/tmp_files/2301.04309v1.pdf.txt b/XtE3T4oBgHgl3EQfFwnp/content/tmp_files/2301.04309v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..58e6cd6e9bf489942848100b60ee51109c39be13 --- /dev/null +++ b/XtE3T4oBgHgl3EQfFwnp/content/tmp_files/2301.04309v1.pdf.txt @@ -0,0 +1,548 @@ +Study of Mach reflection in inviscid flows +S R Siva Prasad Kochi1 +siva.ksr@gmail.com +and +M Ramakrishna2 +krishna@ae.iitm.ac.in +1Dept. of Aerospace Engg., IIT Madras. +2Professor, Dept. of Aerospace Engg., IIT Madras. +January 12, 2023 +Abstract +In this paper, we study the Mach reflection phenomenon in inviscid flows using a higher order +discontinuous Galerkin method and overset grids. We use the shock capturing procedure proposed +in [1] using overset grids to capture the discontinuities occurring in the supersonic flow over a wedge +accurately. In this procedure, we obtain a coarse grid solution first and using the troubled cell data, +we construct an overset grid which is approximately aligned to all the discontinuities. We rerun the +solver with the coarse grid solution as the initial condition while using the troubled cell indicator and +the limiter only on the overset grid. This allows us to capture the discontinuities accurately. Using +this procedure, we have obtained the solution for Mach 3.0 and 4.0 flow over a wedge for various +wedge angles and determined the detachment criterion and the Von Neumann condition accurately. +We have also determined the Mach stem height for various wedge angles for these Mach numbers. +We have also demonstrated the hysteresis that occurs in the transition from regular reflection to +Mach reflection. +Keywords: Mach reflection, discontinuous Galerkin method, overset grids, hysteresis +Paper under review in Shock Waves journal +1 +Introduction +In this paper, we study the transition between regular reflection (RR) and Mach reflection (MR) of steady +shock waves in inviscid flows using discontinuous Galerkin method (DGM) along with overset grids. This +has been studied quite extensively in literature experimentally [2], numerically [3], and analytically [4]. +We use the shock capturing procedure proposed in [1] using overset grids and a higher order method +(DGM) to capture the discontinuities occurring in the supersonic flow over a wedge accurately. In this +procedure, we obtain a coarse grid solution first and using the solution and the troubled cell data, we +construct an overset grid which is approximately aligned to all the discontinuities. We rerun the solver +with the coarse grid solution as the initial condition while using the troubled cell indicator and the +limiter only on the overset grid. This allows us to capture the discontinuities accurately. We solve the +Euler equations in the computational domain shown in Figure 1 using DGM and overset grids for Mach +3.0 and 4.0 flow over a wedge for different wedge angles to determine the transition between RR and +MR. In this way, we determine the detachment criterion and the Von Neumann condition accurately to +demonstrate the hysteresis that occurs in such a flow. +The paper is organized as follows. We describe the formulation of the discontinuous Galerkin method +used for all our results in Section 2, the procedure used for shock capturing using overset grids is described +in Section 3, the results are described in Section 4 and we conclude the paper in Section 5. +2 +Description of discontinuous Galerkin method +Consider the Euler equations in conservative form as given by +∂Q +∂t + ∂F(Q) +∂x ++ ∂G(Q) +∂y += 0 +in the domain +Ω +(1) +where Q = (ρ, ρu, ρv, E)T , F(Q) = uQ + (0, p, 0, pu)T and G(Q) = vQ + (0, 0, p, pv)T with p = +(γ − 1)(E − 1 +2ρ(u2 + v2)) and γ = 1.4. Here, ρ is the density, (u, v) is the velocity, E is the total energy +and p is the pressure. We approximate the domain Ω by K non overlapping elements given by Ωk. +1 +arXiv:2301.04309v1 [math.NA] 11 Jan 2023 + +Hm +w +H +Supersonic +Outflow +Wall +Inflow +Symmetry +Plane +y +x +θw +Figure 1: Computational domain for Mach reflection showing the shock structure and the Mach stem +height (Hm) +We look at solving (1) using the discontinuous Galerkin method. We approximate the local solution +in an element Ωk, where k is the element number, as a polynomial of order N which is given by: +Qk +h(r, s) = +Np−1 +� +i=0 +Qk +i ψi(r, s) +(2) +where Np = (N + 1)(N + 1) and r and s are the local coordinates. Here, the subscript i represents the +particular degree of freedom, h represents the grid size, and the superscript k is the element number. The +polynomial basis used (ψi(r, s)) is the tensor product orthonormalized Legendre polynomials of degree +N. The number of degrees of freedom are given by Np = (N + 1)(N + 1). Now, using ψj(r, s) as the test +function, the weak form of the equation (1) is obtained as +Np−1 +� +i=0 +∂Qk +i +∂t +� +Ωk +ψiψjdΩ + +� +∂Ωk +ˆFψjds − +� +Ωk +⃗F · ∇ψjdΩ = 0 +j = 0, . . . , Np − 1 +(3) +where ∂Ωk is the boundary of Ωk, ⃗F = (F(Q), G(Q)) and ˆF = ¯ +F ∗ · ˆn where ¯ +F ∗ is the monotone numer- +ical flux at the interface which is calculated using an exact or approximate Riemann solver and ˆn is the +unit outward normal. This is termed to be PN based discontinuous Galerkin method. +Equation (3) is integrated using an appropriate Gauss Legendre quadrature and is discretized in time by +using the fifth order Runge-Kutta time discretization given in [5] unless otherwise specified. To control +spurious oscillations which occur near discontinuities, a limiter is used with a troubled cell indicator. We +have used the KXRCF troubled cell indicator [6] and the compact subcell WENO (CSWENO) limiter +proposed in [7] for all our calculations. +2 + +3 +Overset grids and shock capturing +Overset grids consist of multiple grids which overlap each other as shown in Figure 2. When using DGM +on overset grids, there are two possible approaches to handle data communication between the grids. +One is a face based communication approach developed in [8], where solutions at an overset interface +are obtained from the donor element, and then the boundary condition is applied weakly by imposing a +numerical flux at the flux interpolation points. The other is an element based communication approach +developed in [9], where the internal degrees of freedom of cells near the overset interface are obtained +from the donor element. We use the new element based communication approach developed in [10] for +the data communication between the grids. +Grid 1 +Grid 2 +Figure 2: Two overlapping grids (Grid 1 in black and Grid 2 in Red) +For capturing the shocks accurately, we use the procedure developed in [1] using overset grids. A brief +explanation of the procedure is given below: +Step 1: Run the solver on a coarse grid with a given troubled cell indicator and limiter to steady +state and obtain the solution. As an example, we show the coarse grid solution obtained for Mach 3.0 +flow over a 24◦ wedge in Figure 3 +Step 2: Look at the troubled cells to locate the discontinuities (shocks) that occur in the solution. +The troubled cell profile obtained for the Mach 3.0 flow over a 24◦ wedge using the KXRCF troubled +cell indicator [6] is shown in Figure 4. From this figure, we can see that the troubled cells give us a good +idea of the location of the shocks, the contact discontinuity and the initial expansion that forms near the +wedge. +Step 3: Construct an overset grid conforming to the computational domain which is refined in a di- +rection perpendicular to the discontinuities such that they are approximately parallel to a grid line. +This overset grid also encompasses all the troubled cells. An example overset grid constructed in such a +fashion for the Mach 3.0 flow over a 24◦ wedge is shown in Figure 5. +Step 4: Using this overset grid, we rerun the solver with the coarse grid solution as the initial condition. +While running the solver, we use the troubled cell indicator and the limiter only on the overset grid. We +also use a high resolution numerical flux on the overset grid to capture the shock accurately. We have +used the SLAU2 [11] numerical flux in the overset grid and the less expensive Lax-Friedrichs flux else- +where. Using this procedure, we obtain a more accurate solution with the discontinuities approximately +aligned to a grid line. The final solution obtained in this fashion for Mach 3.0 flow over a 24◦ wedge is +shown in Figure 6. +3 + +Figure 3: Coarse grid solution for Mach 3.0 flow over a 24◦ wedge using P1 based discontinuous Galerkin +method +4 + +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1+ +0.9 +0.9 +0.8- +0.8 +0.7- +0.7 +0.6 +0.6 +3.1 +0.5- +0.5 +2.5 +Number +0.4 +0.4 +0.3 +0.3 +2 +1.5 +Mach +0.2 +0.2 +1 +0.1- +0.1 +0.4 +LO +0 +0.2 +0.4 +0.6 +0.8 +1.2 +1.4 +1.6Figure 4: Troubled cells obtained using the KXRCF troubled cell indicator [6] for Mach 3.0 flow over a +24◦ wedge +5 + +Figure 5: Example overset grid for Mach 3.0 flow over a 24◦ wedge constructed such that the disconti- +nuities are approximately parallel to a grid line +6 + +LYFigure 6: Final solution obtained for Mach 3.0 flow over a 24◦ wedge using an overset grid and P4 based +discontinuous Galerkin method with discontinuities captured accurately +Regarding the computational cost of the scheme proposed for the current problem, we note that the +troubled cell data shown in Figure 4 are obtained from a coarse grid of 20000 elements using P1 based +DGM after converging to a residue of about 1 × 10−6 which happens in about 8000 iterations. This +coarse grid solution is not a fully converged solution but is good enough to be used as an initial condition +for the overset grid higher order (P4 based DGM) solution. The time taken per iteration per degree of +freedom using P1 based DGM for this problem is about 1.084 × 10−6s. This tells us that the coarse grid +troubled cell data is obtained in about 693.76s which is a little less than 12 minutes. Most of the work +involved is in constructing the overset grid which is done manually. After the overset grid is constructed +and the solver is run on the new grid with the coarse grid solution as the initial condition, the residue +converges to 1 × 10−16 in about 32000 iterations. For P4 based DGM, the time taken per iteration per +degree of freedom is about 1.345 × 10−6s for this problem. This tells us that the final solution (which is +fifth order accurate) is obtained in about 21520s which is about six hours. All these calculations are done +on a 3.60 GHz, Intel(R) Core(TM) i7-7700 CPU with a single thread. The calculations for remaining +angles are quite similar. We also note that we have a parallelised code and we obtain the solution much +faster based on the number of threads used. +4 +Results +1) Mach number 3.0: We solve the two-dimensional Euler equations given by (1) using the discon- +tinuous Galerkin method for various wedge angles between θw = 19.5◦ and θw = 24◦ near the transition +criterion (transition from Regular reflection to Mach reflection) for Mach number 3.0 and w/H = 1.0 +in the computational domain shown in Figure 1. We consider two cases to demonstrate the hysteresis +phenomenon. We solve the equations using an impulsive start as the initial condition for the first case, +and the converged solution for θw = 24◦ as the second case. The first case is a numerical model to +7 + +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1+ +X +X +0.9 +0.9 +0.8- +0.8 +0.7- +0.7 +3.0 +0.6- +0.6 +2.5 +0.5 +-0.5 +Mach Number +2 +0.4- +0.4 +0.3- +1.5 +-0.3 +0.2- +-0.2 +1 +0.1 +-0.1 +0.4 +LO +0 +0 +0.2 +0.4 +0.6 +0.8 +1.2 +1.4 +1.6obtain the detachment criterion and the second case demonstrates the Von Neumann condition. Using +the procedure outlined in Section 3, we obtain a very accurate solution for each of the wedge angles +and obtain the transition criterion clearly. Note that the initial conditions mentioned above are used to +obtain the coarse grid solution first and the coarse grid solution is used as an initial condition to get +the fine grid solution using P4 based DGM. To show that the transition criterion has been captured +accurately, we show the solution obtained for Mach 3.0 flow over a wedge with wedge angles θw = 21.45◦ +and θw = 21.46◦ with the case 1 initial conditions in Figures 7 and 8 respectively. +Figure 7: Final solution obtained for Mach 3.0 flow over a 21.45◦ wedge with an overset grid and P4 +based discontinuous Galerkin method with discontinuities captured accurately using an impulsive start +We also show the table containing the Mach stem height obtained using this procedure for various wedge +angles for both cases of initial conditions in Table 1. +From Table 1, we can see that the transition +criterion occurs between wedge angles 21.45◦ and 21.46◦ and the Von Neumann transition criterion +occurs between wedge angles 19.6◦ and 19.7◦. From the three-shock theory, the transition criterion is at +θw = 21.456◦ and the Von Neumann condition occurs at θw = 19.656◦. This closely agrees with what +we have obtained and validates our procedure. +2) Mach number 4.0: We repeat the same procedure for various wedge angles between θw = 20.5◦ and +θw = 27◦ near the transition criterion (transition from Regular reflection to Mach reflection) for Mach +number 4.0 and w/H = 1.0 in the computational domain shown in Figure 1. Again, we consider two +cases to demonstrate the hysteresis phenomenon. We solve the equations using an impulsive start as the +initial condition for the first case, and the converged solution for θw = 27◦ as the second case. We show +the table containing the Mach stem height obtained using this procedure for various wedge angles for +both cases of initial conditions in Table 2. From Table 2, we can see that the transition criterion occurs +between wedge angles 25.6◦ and 25.7◦ and the Von Neumann transition criterion occurs between wedge +angles 20.9◦ and 20.8◦. From the three-shock theory, the transition criterion for Mach number 4.0 is at +θw = 25.61◦ and the Von Neumann condition occurs at θw = 20.86◦. This closely agrees with what we +have obtained and validates our procedure. +8 + +0 +0.2 +0.4 +0.6 +0.8 +1.2 +1.4 +1.6 +1.8 +1- +0.9 +0.9 +0.8 +0.8 +0.7 +0.7 +3.0 +0.6 +0.6 +Mach Number +2.5 +0.5 +0.5 +0.4 +-0.4 +2 +0.3- +0.3 +1.5 +0.2- +-0.2 +0.1 +-0.1 +0.9 +0 +0 +0.2 +0.4 +0.6 +0.8 +1.2 +1.4 +1.6 +1.8Figure 8: Final solution obtained for Mach 3.0 flow over a 21.46◦ wedge with an overset grid and P4 +based discontinuous Galerkin method with discontinuities captured accurately using an impulsive start +9 + +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +0.9 +0.9 +0.8- +0.8 +0.7 +-0.7 +0.6 +0.6 +3.0 +0.5 +0.5 +2.5 +Number +0.4 +-0.4 +2 +0.3- +0.3 +1.5 +0.2 +0.2 +Mach +0.1- +0.1 +0 +0 +0.5 +0 +0.2 +0.4 +0.6 +0.8 +1.2 +1.4 +1.6 +1.8Table 1: Non-dimensionalised height of the Mach stem (Hm/H) as a function of wedge angle (θw) for +M=3.0 and w/H = 1.0 using two sets of initial conditions:- Case 1: Impulsive start, Case 2: Converged +solution for θw = 24◦ +θw +Hm/H for case 1 +Hm/H for case 2 +(impulsive start) +(converged solution for θw = +24◦) +24◦ +0.274 +- +23.5◦ +0.233 +0.233 +23◦ +0.188 +0.188 +22.5◦ +0.153 +0.153 +22◦ +0.122 +0.122 +21.46◦ +0.078 +0.078 +21.45◦ +RR +0.066 +21◦ +RR +0.041 +20.5◦ +RR +0.028 +20◦ +RR +0.009 +19.7◦ +RR +0.002 +19.6◦ +RR +RR +19.5◦ +RR +RR +Table 2: Non-dimensionalised height of the Mach stem (Hm/H) as a function of wedge angle (θw) for +M=4.0 and w/H = 1.0 using two sets of initial conditions:- Case 1: Impulsive start, Case 2: Converged +solution for θw = 27◦ +θw +Hm/H for case 1 +Hm/H for case 2 +(impulsive start) +(converged solution for θw = +27◦) +27◦ +0.344 +- +26.5◦ +0.313 +0.313 +26◦ +0.285 +0.285 +25.7◦ +0.245 +0.245 +25.6◦ +RR +0.224 +25.5◦ +RR +0.203 +25◦ +RR +0.183 +24.5◦ +RR +0.144 +24◦ +RR +0.115 +23◦ +RR +0.093 +22◦ +RR +0.045 +21◦ +RR +0.005 +20.9◦ +RR +0.002 +20.8◦ +RR +RR +20.5◦ +RR +RR +5 +Conclusion +We have demonstrated the transition between regular reflection (RR) and Mach reflection (MR) of steady +shock waves in inviscid flows using discontinuous Galerkin method (DGM) along with overset grids using +an accurate shock capturing procedure [1]. We have identified the detachment criterion and the Von +Neumann condition accurately and demonstrated the hysteresis which occurs in the transition. We have +also calculated the Mach stem height for various wedge angles for Mach numbers 3.0 and 4.0. As future +work, this shock capturing procedure can be used to further study the different flow phenomena that +occur when the inflow Mach number is small (< 2.0). +10 + +References +[1] Siva Prasad Kochi S.R. and Ramakrishna M., “Shock capturing with discontinuous Galerkin Method +using Overset grids for two-dimensional Euler equations,” 2020. Paper under review in J. Comput. +Phys. Preprint at https://arxiv.org/abs/2003.01378. +[2] Chpoun A., Passerel D., Li H. and Ben-Dor G., “Reconsideration of the state-of-the-art of oblique +shock wave reflection in steady flows. Part I: Experimental investigation,” J. Fluid. Mech., vol. 301, +pp. 19–35, 1995. +[3] Vuillon J., Zeitoun D. and Ben-Dor G., “Reconsideration of the state-of-the-art of oblique shock +wave reflection in steady flows. Part 2: Numerical investigation,” J. Fluid. Mech., vol. 301, pp. 37–50, +1995. +[4] Ben-Dor G., Shock wave reflection phenomena. New York: Springer, 2007. +[5] Butcher J.C., Numerical Methods for Ordinary Differential Equations. 3rd edition: John Wiley and +Sons, 2016. +[6] Krivodonova L., Xin J., Remacle J.-F., Chevaugeon N. and Flaherty J., “Shock detection and lim- +iting with discontinuous Galerkin methods for hyperbolic conservation laws,” Appl. Numer. Math., +vol. 48, pp. 323–338, 2004. +[7] Siva Prasad Kochi S.R. and Ramakrishna M., “A compact subcell WENO limiting strategy using +immediate neighbours for Runge-Kutta discontinuous Galerkin methods,” Int. J. Comput. Math., +vol. 98(3), pp. 608–626, 2021. +[8] Galbraith M.C., Benek J.A., Orkwis P.D. and Turner M.G., “A Discontinuous Galerkin Chimera +scheme,” Computers and Fluids, vol. 98, pp. 27–53, 2014. +[9] Nastase C.R., Mavriplis D.J. and Sitaraman J., “An Overset Unstructured Mesh Discontinuous +Galerkin Approach for Aerodynamic Problems,” AIAA 2011-195, 2011. +[10] Siva Prasad Kochi S.R. and Ramakrishna M., “A Discontinuous Galerkin Overset Scheme Using +WENO Reconstruction and Subcells for Two-Dimensional Problems,” J. Sci. Comput., vol. 93:35, +2022. +[11] Kitamura K. and Shima E., “Towards shock-stable and accurate hypersonic heating computations: +A new pressure flux for AUSM-family schemes,” J. Comput. Phys., vol. 245, pp. 62–83, 2013. +11 + diff --git a/XtE3T4oBgHgl3EQfFwnp/content/tmp_files/load_file.txt b/XtE3T4oBgHgl3EQfFwnp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..479cb7eefed5d43fe2fb9c8c271f1f9b8a2e2ccc --- /dev/null +++ b/XtE3T4oBgHgl3EQfFwnp/content/tmp_files/load_file.txt @@ -0,0 +1,442 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf,len=441 +page_content='Study of Mach reflection in inviscid flows S R Siva Prasad Kochi1 siva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='ksr@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='com and M Ramakrishna2 krishna@ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='iitm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='in 1Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' of Aerospace Engg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=', IIT Madras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' 2Professor, Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' of Aerospace Engg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=', IIT Madras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' January 12, 2023 Abstract In this paper, we study the Mach reflection phenomenon in inviscid flows using a higher order discontinuous Galerkin method and overset grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We use the shock capturing procedure proposed in [1] using overset grids to capture the discontinuities occurring in the supersonic flow over a wedge accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' In this procedure, we obtain a coarse grid solution first and using the troubled cell data, we construct an overset grid which is approximately aligned to all the discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We rerun the solver with the coarse grid solution as the initial condition while using the troubled cell indicator and the limiter only on the overset grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' This allows us to capture the discontinuities accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Using this procedure, we have obtained the solution for Mach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 flow over a wedge for various wedge angles and determined the detachment criterion and the Von Neumann condition accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We have also determined the Mach stem height for various wedge angles for these Mach numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We have also demonstrated the hysteresis that occurs in the transition from regular reflection to Mach reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Keywords: Mach reflection, discontinuous Galerkin method, overset grids, hysteresis Paper under review in Shock Waves journal 1 Introduction In this paper, we study the transition between regular reflection (RR) and Mach reflection (MR) of steady shock waves in inviscid flows using discontinuous Galerkin method (DGM) along with overset grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' This has been studied quite extensively in literature experimentally [2], numerically [3], and analytically [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We use the shock capturing procedure proposed in [1] using overset grids and a higher order method (DGM) to capture the discontinuities occurring in the supersonic flow over a wedge accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' In this procedure, we obtain a coarse grid solution first and using the solution and the troubled cell data, we construct an overset grid which is approximately aligned to all the discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We rerun the solver with the coarse grid solution as the initial condition while using the troubled cell indicator and the limiter only on the overset grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' This allows us to capture the discontinuities accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We solve the Euler equations in the computational domain shown in Figure 1 using DGM and overset grids for Mach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 flow over a wedge for different wedge angles to determine the transition between RR and MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' In this way, we determine the detachment criterion and the Von Neumann condition accurately to demonstrate the hysteresis that occurs in such a flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We describe the formulation of the discontinuous Galerkin method used for all our results in Section 2, the procedure used for shock capturing using overset grids is described in Section 3, the results are described in Section 4 and we conclude the paper in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' 2 Description of discontinuous Galerkin method Consider the Euler equations in conservative form as given by ∂Q ∂t + ∂F(Q) ∂x + ∂G(Q) ∂y = 0 in the domain Ω (1) where Q = (ρ, ρu, ρv, E)T , F(Q) = uQ + (0, p, 0, pu)T and G(Q) = vQ + (0, 0, p, pv)T with p = (γ − 1)(E − 1 2ρ(u2 + v2)) and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Here, ρ is the density, (u, v) is the velocity, E is the total energy and p is the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We approximate the domain Ω by K non overlapping elements given by Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='04309v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='NA] 11 Jan 2023 Hm w H Supersonic Outflow Wall Inflow Symmetry Plane y x θw Figure 1: Computational domain for Mach reflection showing the shock structure and the Mach stem height (Hm) We look at solving (1) using the discontinuous Galerkin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We approximate the local solution in an element Ωk, where k is the element number, as a polynomial of order N which is given by: Qk h(r, s) = Np−1 � i=0 Qk i ψi(r, s) (2) where Np = (N + 1)(N + 1) and r and s are the local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Here, the subscript i represents the particular degree of freedom, h represents the grid size, and the superscript k is the element number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' The polynomial basis used (ψi(r, s)) is the tensor product orthonormalized Legendre polynomials of degree N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' The number of degrees of freedom are given by Np = (N + 1)(N + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Now, using ψj(r, s) as the test function, the weak form of the equation (1) is obtained as Np−1 � i=0 ∂Qk i ∂t � Ωk ψiψjdΩ + � ∂Ωk ˆFψjds − � Ωk ⃗F · ∇ψjdΩ = 0 j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' , Np − 1 (3) where ∂Ωk is the boundary of Ωk, ⃗F = (F(Q), G(Q)) and ˆF = ¯ F ∗ · ˆn where ¯ F ∗ is the monotone numer- ical flux at the interface which is calculated using an exact or approximate Riemann solver and ˆn is the unit outward normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' This is termed to be PN based discontinuous Galerkin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Equation (3) is integrated using an appropriate Gauss Legendre quadrature and is discretized in time by using the fifth order Runge-Kutta time discretization given in [5] unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' To control spurious oscillations which occur near discontinuities, a limiter is used with a troubled cell indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We have used the KXRCF troubled cell indicator [6] and the compact subcell WENO (CSWENO) limiter proposed in [7] for all our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' 2 3 Overset grids and shock capturing Overset grids consist of multiple grids which overlap each other as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' When using DGM on overset grids, there are two possible approaches to handle data communication between the grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' One is a face based communication approach developed in [8], where solutions at an overset interface are obtained from the donor element, and then the boundary condition is applied weakly by imposing a numerical flux at the flux interpolation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' The other is an element based communication approach developed in [9], where the internal degrees of freedom of cells near the overset interface are obtained from the donor element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We use the new element based communication approach developed in [10] for the data communication between the grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Grid 1 Grid 2 Figure 2: Two overlapping grids (Grid 1 in black and Grid 2 in Red) For capturing the shocks accurately, we use the procedure developed in [1] using overset grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' A brief explanation of the procedure is given below: Step 1: Run the solver on a coarse grid with a given troubled cell indicator and limiter to steady state and obtain the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' As an example, we show the coarse grid solution obtained for Mach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 flow over a 24◦ wedge in Figure 3 Step 2: Look at the troubled cells to locate the discontinuities (shocks) that occur in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' The troubled cell profile obtained for the Mach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 flow over a 24◦ wedge using the KXRCF troubled cell indicator [6] is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' From this figure, we can see that the troubled cells give us a good idea of the location of the shocks, the contact discontinuity and the initial expansion that forms near the wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Step 3: Construct an overset grid conforming to the computational domain which is refined in a di- rection perpendicular to the discontinuities such that they are approximately parallel to a grid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' This overset grid also encompasses all the troubled cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' An example overset grid constructed in such a fashion for the Mach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 flow over a 24◦ wedge is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Step 4: Using this overset grid, we rerun the solver with the coarse grid solution as the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' While running the solver, we use the troubled cell indicator and the limiter only on the overset grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We also use a high resolution numerical flux on the overset grid to capture the shock accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We have used the SLAU2 [11] numerical flux in the overset grid and the less expensive Lax-Friedrichs flux else- where.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Using this procedure, we obtain a more accurate solution with the discontinuities approximately aligned to a grid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' The final solution obtained in this fashion for Mach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 flow over a 24◦ wedge is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' 3 Figure 3: Coarse grid solution for Mach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 flow over a 24◦ wedge using P1 based discontinuous Galerkin method 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 1+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='7- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 Number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='3 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 Mach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='1- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 LO 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6Figure 4: Troubled cells obtained using the KXRCF troubled cell indicator [6] for Mach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 flow over a 24◦ wedge 5 Figure 5: Example overset grid for Mach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 flow over a 24◦ wedge constructed such that the disconti- nuities are approximately parallel to a grid line 6 LYFigure 6: Final solution obtained for Mach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 flow over a 24◦ wedge using an overset grid and P4 based discontinuous Galerkin method with discontinuities captured accurately Regarding the computational cost of the scheme proposed for the current problem, we note that the troubled cell data shown in Figure 4 are obtained from a coarse grid of 20000 elements using P1 based DGM after converging to a residue of about 1 × 10−6 which happens in about 8000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' This coarse grid solution is not a fully converged solution but is good enough to be used as an initial condition for the overset grid higher order (P4 based DGM) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' The time taken per iteration per degree of freedom using P1 based DGM for this problem is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='084 × 10−6s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' This tells us that the coarse grid troubled cell data is obtained in about 693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='76s which is a little less than 12 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Most of the work involved is in constructing the overset grid which is done manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' After the overset grid is constructed and the solver is run on the new grid with the coarse grid solution as the initial condition, the residue converges to 1 × 10−16 in about 32000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' For P4 based DGM, the time taken per iteration per degree of freedom is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='345 × 10−6s for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' This tells us that the final solution (which is fifth order accurate) is obtained in about 21520s which is about six hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' All these calculations are done on a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='60 GHz, Intel(R) Core(TM) i7-7700 CPU with a single thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' The calculations for remaining angles are quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We also note that we have a parallelised code and we obtain the solution much faster based on the number of threads used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' 4 Results 1) Mach number 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0: We solve the two-dimensional Euler equations given by (1) using the discon- tinuous Galerkin method for various wedge angles between θw = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5◦ and θw = 24◦ near the transition criterion (transition from Regular reflection to Mach reflection) for Mach number 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 and w/H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 in the computational domain shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We consider two cases to demonstrate the hysteresis phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We solve the equations using an impulsive start as the initial condition for the first case, and the converged solution for θw = 24◦ as the second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' The first case is a numerical model to 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 1+ X X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='7- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 Mach Number 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='3- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 LO 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6obtain the detachment criterion and the second case demonstrates the Von Neumann condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Using the procedure outlined in Section 3, we obtain a very accurate solution for each of the wedge angles and obtain the transition criterion clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Note that the initial conditions mentioned above are used to obtain the coarse grid solution first and the coarse grid solution is used as an initial condition to get the fine grid solution using P4 based DGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' To show that the transition criterion has been captured accurately, we show the solution obtained for Mach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 flow over a wedge with wedge angles θw = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='45◦ and θw = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='46◦ with the case 1 initial conditions in Figures 7 and 8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Figure 7: Final solution obtained for Mach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 flow over a 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='45◦ wedge with an overset grid and P4 based discontinuous Galerkin method with discontinuities captured accurately using an impulsive start We also show the table containing the Mach stem height obtained using this procedure for various wedge angles for both cases of initial conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' From Table 1, we can see that the transition criterion occurs between wedge angles 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='45◦ and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='46◦ and the Von Neumann transition criterion occurs between wedge angles 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6◦ and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='7◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' From the three-shock theory, the transition criterion is at θw = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='456◦ and the Von Neumann condition occurs at θw = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='656◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' This closely agrees with what we have obtained and validates our procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' 2) Mach number 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0: We repeat the same procedure for various wedge angles between θw = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5◦ and θw = 27◦ near the transition criterion (transition from Regular reflection to Mach reflection) for Mach number 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 and w/H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 in the computational domain shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Again, we consider two cases to demonstrate the hysteresis phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We solve the equations using an impulsive start as the initial condition for the first case, and the converged solution for θw = 27◦ as the second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We show the table containing the Mach stem height obtained using this procedure for various wedge angles for both cases of initial conditions in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' From Table 2, we can see that the transition criterion occurs between wedge angles 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6◦ and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='7◦ and the Von Neumann transition criterion occurs between wedge angles 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='9◦ and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' From the three-shock theory, the transition criterion for Mach number 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 is at θw = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='61◦ and the Von Neumann condition occurs at θw = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='86◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' This closely agrees with what we have obtained and validates our procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8 1- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 Mach Number 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='3- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='9 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8Figure 8: Final solution obtained for Mach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 flow over a 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='46◦ wedge with an overset grid and P4 based discontinuous Galerkin method with discontinuities captured accurately using an impulsive start 9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 Number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8Table 1: Non-dimensionalised height of the Mach stem (Hm/H) as a function of wedge angle (θw) for M=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 and w/H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 using two sets of initial conditions:- Case 1: Impulsive start, Case 2: Converged solution for θw = 24◦ θw Hm/H for case 1 Hm/H for case 2 (impulsive start) (converged solution for θw = 24◦) 24◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='274 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='233 23◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='188 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='153 22◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='122 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='46◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='078 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='45◦ RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='066 21◦ RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='041 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5◦ RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='028 20◦ RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='009 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='7◦ RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='002 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6◦ RR RR 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5◦ RR RR Table 2: Non-dimensionalised height of the Mach stem (Hm/H) as a function of wedge angle (θw) for M=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 and w/H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 using two sets of initial conditions:- Case 1: Impulsive start, Case 2: Converged solution for θw = 27◦ θw Hm/H for case 1 Hm/H for case 2 (impulsive start) (converged solution for θw = 27◦) 27◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='344 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='313 26◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='285 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='7◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='245 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='6◦ RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='224 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5◦ RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='203 25◦ RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='183 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5◦ RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='144 24◦ RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='115 23◦ RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='093 22◦ RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='045 21◦ RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='005 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='9◦ RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='002 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='8◦ RR RR 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='5◦ RR RR 5 Conclusion We have demonstrated the transition between regular reflection (RR) and Mach reflection (MR) of steady shock waves in inviscid flows using discontinuous Galerkin method (DGM) along with overset grids using an accurate shock capturing procedure [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We have identified the detachment criterion and the Von Neumann condition accurately and demonstrated the hysteresis which occurs in the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' We have also calculated the Mach stem height for various wedge angles for Mach numbers 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' As future work, this shock capturing procedure can be used to further study the different flow phenomena that occur when the inflow Mach number is small (< 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' 10 References [1] Siva Prasad Kochi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' and Ramakrishna M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=', “Shock capturing with discontinuous Galerkin Method using Overset grids for two-dimensional Euler equations,” 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Paper under review in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Preprint at https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='org/abs/2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content='01378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' [2] Chpoun A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=', Passerel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=', Li H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' and Ben-Dor G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=', “Reconsideration of the state-of-the-art of oblique shock wave reflection in steady flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Part I: Experimental investigation,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' 301, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' 19–35, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' [3] Vuillon J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=', Zeitoun D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' and Ben-Dor G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=', “Reconsideration of the state-of-the-art of oblique shock wave reflection in steady flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfFwnp/content/2301.04309v1.pdf'} +page_content=' Part 2: Numerical investigation,” J.' metadata={'source': 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0000000000000000000000000000000000000000..61a1e72ddf67e28c7eeeb3989f766ae7ec45a2e8 --- /dev/null +++ b/YNFPT4oBgHgl3EQftDXh/content/tmp_files/2301.13151v1.pdf.txt @@ -0,0 +1,1732 @@ +Original Paper +Convolutional Neural Network–Based Automatic Classification of +Colorectal and Prostate Tumor Biopsies Using Multispectral +Imagery: System Development Study +Remy Peyret1, PhD; Duaa alSaeed2, PhD; Fouad Khelifi1, PhD; Nadia Al-Ghreimil2, PhD; Heyam Al-Baity2, PhD; +Ahmed Bouridane1, PhD +1Northumbria University at Newcastle, Newcastle, United Kingdom +2College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia +Corresponding Author: +Duaa alSaeed, PhD +College of Computer and Information Sciences +King Saud University +King Abdullah Road +Riyadh, 11451 +Saudi Arabia +Phone: 966 555442477 +Email: dalsaeed@ksu.edu.sa +Abstract +Background: Colorectal and prostate cancers are the most common types of cancer in men worldwide. To diagnose colorectal +and prostate cancer, a pathologist performs a histological analysis on needle biopsy samples. This manual process is time-consuming +and error-prone, resulting in high intra- and interobserver variability, which affects diagnosis reliability. +Objective: This study aims to develop an automatic computerized system for diagnosing colorectal and prostate tumors by +using images of biopsy samples to reduce time and diagnosis error rates associated with human analysis. +Methods: In this study, we proposed a convolutional neural network (CNN) model for classifying colorectal and prostate tumors +from multispectral images of biopsy samples. The key idea was to remove the last block of the convolutional layers and halve +the number of filters per layer. +Results: Our results showed excellent performance, with an average test accuracy of 99.8% and 99.5% for the prostate and +colorectal data sets, respectively. The system showed excellent performance when compared with pretrained CNNs and other +classification methods, as it avoids the preprocessing phase while using a single CNN model for the whole classification task. +Overall, the proposed CNN architecture was globally the best-performing system for classifying colorectal and prostate tumor +images. +Conclusions: The proposed CNN architecture was detailed and compared with previously trained network models used as +feature extractors. These CNNs were also compared with other classification techniques. As opposed to pretrained CNNs and +other classification approaches, the proposed CNN yielded excellent results. The computational complexity of the CNNs was +also investigated, and it was shown that the proposed CNN is better at classifying images than pretrained networks because it +does not require preprocessing. Thus, the overall analysis was that the proposed CNN architecture was globally the best-performing +system for classifying colorectal and prostate tumor images. +(JMIR Bioinform Biotech 2022;3(1):e27394) doi: 10.2196/27394 +KEYWORDS +convolutional neural networks; classification; colorectal tumor; prostate tumor; machine learning; image processing +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 1 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + +Introduction +Background +According to the World Health Organization 2014 report, 14 +million new cases of cancer were diagnosed in 2012, and the +disease caused 8 million people to die in the same period [1]. +Colorectal cancer is the third most common cancer globally, +whereas prostate cancer is the second most common cancer +among men, accounting for 9.7% and 7.9% of all cancers in +both sexes, respectively [1]. Both colorectal and prostate tissues +are glandular and therefore have a similar histological +appearance. +For prostate cancer diagnosis, the European Association of +Urology guidelines [2] recommend the performance of a +histological analysis on a sample taken from a needle biopsy +by a pathologist who decides the grade and stage of cancer or +the type of tumor based on their experience and expertise. +However, this process is time consuming and it also results in +a high intra- and interobserver variability [3,4], which affects +diagnosis reliability. In December 1999, a study [5] of more +than 6000 patients conducted by Johns Hopkins researchers +found that up to 2 out of every 100 people who came to larger +medical centers for treatment were given an incorrect diagnosis +after histological analysis. These results suggest that +second-opinion pathology examinations not only prevent errors +but also save lives and money. Consequently, there is an +increasing interest among pathology experts in the use of +machine vision (or computational diagnosis tools) to reduce +diagnosis error rates by lowering the fallible aspect of human +image interpretation. +Computer-aided diagnosis can assist pathologists in reducing +the human analysis time, improving efficiency, and acting as a +second opinion [6-8]. Adding computer-based quantitative +analysis to human qualitative interpretation could significantly +reduce the intra- and interobserver variability revealed in [4]. +The main objective of this study is to develop an automatic +computerized system for the diagnosis of colorectal and prostate +tumors using images of biopsy samples. +Numerous investigations concerning prostate or colorectal tumor +classification have been carried out [9,10]. However, most use +color spaces limited to gray-scale or red, green, blue (RGB) +images. In the last decade, many studies have used multispectral +images [11-18], which are acquired using a more precise +sampling of the light spectrum. This approach aims to better +capture the spectrum of the reflected light coming from the +observed sample, offering more discriminative information. +Lasch et al [19] suggested that multispectral imagery can +improve histopathological analysis by capturing patterns that +are invisible to the human vision system and standard RGB +imaging. Multispectral imaging studies have shown promising +results and often outperformed systems using traditional +gray-scale or RGB images [9,10]. However, multispectral +images contain a large amount of data, making them more +difficult to process because of increased execution time and +problems caused by the curse of dimensionality [13]. +Since the emergence of graphic processing units (GPUs) with +sufficient processing power to train Convolutional neural +networks (CNNs) in 2011, these models have seen a growing +interest in image classification. Several models have been +developed and tested on the ImageNet data set. As an example, +the AlexNet architecture was developed in 2012 [20] and won +several international competitions, including the ImageNet +competition. GoogLeNet [21], a 22 layers deep network, won +the ImageNet competition of 2014. He et al [22] deepened the +networks even more with ResNet and won the best paper in +2015 at the Conference on Computer Vision and Pattern +Recognition. To reduce training times, they developed a +framework in which layers are formulated as a residual function +with reference to the layer input, as opposed to the unreferenced +learning functions previously used. The residual network +comprised 152 layers. In 2016, Google DeepMind used a mix +of supervised deep learning and reinforcement learning (ie, deep +reinforcement learning) to create a system capable of learning +how to play the game of Go [23]. This program, called AlphaGo, +achieved a 99.8% winning rate against other Go programs and +defeated the human European Go champion by 5 games to 0. +In 2017, they created AlphaGo Zero [24], which outperformed +the original AlphaGo in terms of performance and learning time +without using any human knowledge. CNNs seem particularly +adapted to the problem of microscopic images of tumor +classification. A previous study [25] applied CNNs to +microscopic images of colorectal cancer and found a promising +accuracy of 99.1%. However, in this study, images were +preprocessed using an active contour model before being fed +to the CNN model. This operation requires the intervention of +a pathologist to select the region of interest from the segmented +image. Otherwise, this step can be replaced by another +supervised learning model, which requires more training and +thus dramatically increases the processing time. This study +proposes a model that does not require a preprocessing phase +and uses a single CNN model for the entire classification task +using multispectral images. +Deep learning is a branch of machine learning that attempts to +mimic the thinking process. To process data, information is +passed through a network consisting of different layers, where +each layer serves as input to the following layer. The first layer +of a network is referred to as the input layer, whereas the last +layer is the output layer. All the layers in between are called +hidden layers. Typically, a layer is a simple algorithm that +consists of an activation function. This field of machine learning +is now very active, and the research community is focused on +solving practical applications using modern deep learning. This +study aims to apply the deep learning framework to the problem +at hand. +Objective +The primary objective of this study is to develop a computerized +automatic system for the diagnosis of colorectal and prostate +tumors using images of biopsy samples to reduce time and +diagnosis error rates associated with human analysis. To achieve +this, we propose a CNN model for the classification of colorectal +and prostate tumors from multispectral images of biopsy +samples. The key idea is based on removing the last block of +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 2 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + +the convolutional layers and halving the number of filters per +layer. +This paper is organized as follows: we first describe the +principles of deep neural networks. The second section discusses +the proposed method, whereas the data sets of multispectral +tumor images are described in the third section. In the fourth +section, the experiments carried out to validate the approach +are detailed, and finally their results are presented and analyzed. +Feedforward Neural Networks +Overview +Feedforward neural networks, also called multilayer perceptrons +(MLPs), are the basis of deep learning models. They aim to +approximate the function f:~x!y, where ~x is an input feature +vector and y is its corresponding class. The network builds a +mapping ~y=f(~x;) by learning the parameters that provide the +best approximation function to f. In this type of network, +information moves from the input to the output through +intermediate layers with no feedback connections. The number +of layers is called the network depth. Each layer consists of a +vector of functions or units that act in parallel, and the dimension +of this vector is the width of the layer. Therefore, many +hyperparameters need to be chosen when designing a neural +network model, including its architecture, that is, the number +of layers and units per layer. +A hidden layer computes an affine transformation of its input +and then applies a nonlinear function g. This is defined by +h=g(W~x+b), where h is the output of the hidden layer, W is the +weight of the affine transformation, and b is the bias. W and b +are the parameters learned when training the model. +The function chosen for each unit is called the activation +function and is inspired by the behavior of biological neurons. +The most widely used activation function is the rectified linear +unit (ReLU), defined by g(z)=max(0, z). Many other options are +available, and the research on activation function is still a very +active field. However, the ReLU has proven to perform well +and is the default choice for activation functions. +Network training is performed using gradient descent. The main +difference from other models is that the nonlinearity of neural +networks causes the loss function to be nonconvex. Unlike +convex optimization used with support vector machines or deep +reinforcement learning, there is no guarantee of global +convergence of a gradient descent applied to a nonconvex loss +function. Consequently, the learning process is sensitive to the +initial values of weights and biases. To apply gradient-based +learning, a cost function must be chosen. The problem at hand +in this study defines a conditional distribution p(y|x; θ) and the +maximum likelihood principle is well adapted for it [26]. As a +result, the cross-entropy between the training data and the +model’s prediction, which is equivalent to the negative +log-likelihood, is used as the cost function. It enables the model +to estimate the conditional probability of the classes if the input +is known. The cost function model is as follows: +where +is the distribution of the training data and pmodel is +the model distribution and the set of parameters for which the +cost function is calculated. Consequently, the specific form of +the cost function changes depending on the form of the log +pmodel. +Back-Propagation +During training, the gradient of the cost function ΔθJ (θ) is +computed using a back-propagation algorithm [27-29] to allow +information to flow backward through the network and compute +the error made on each network weight. A gradient descent was +then used to minimize the cost function. Learning was +subsequently performed by updating the weights of the units. +This procedure is described in the algorithm shown in Figure +1. +Training a neural network consists of applying a series of +forwarding propagations—the network output is generated from +the data through the network, and back-propagations compute +the error at each unit. Each of these forward propagation and +back-propagation combinations is called a pass. A pass of all +the training examples is performed to compute the gradient used +for the gradient-descent algorithm. A pass of every training +example is called an epoch. At the end of each epoch, the +network weights are updated using a learning rate +hyperparameter, which is multiplied by the gradient calculated +with back-propagation. +The learning rate is one of the most important hyperparameters +for tuning in a neural network, as it controls the effective +capacity of the network [26]. Therefore, it needs to be carefully +optimized. If the learning rate is too large, the gradient descent +can have the opposite of the desired effect, and training accuracy +can decrease [30]. However, when it is too small, the training +is slower, and sometimes the training accuracy can stay +permanently small [30]. The number of epochs is also a +hyperparameter that can be tuned ahead of the training. +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 3 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + +J(O) = -Ex,Y~pdata +log Pmodel(y|x)PdataFigure 1. Back-propagation algorithm. +Methods +Overview +As previously mentioned, the research community is now +focusing on solving practical applications using deep learning +approaches. Our proposed solution to the problem of diagnosing +colorectal and prostate cancer is to apply a deep learning +framework. +CNNs [27,31] are a type of neural network that specialize in +data with a grid-like topology. They are particularly adapted +for image processing. Similar to conventional neural networks, +they consist of units with weights and biases that are learned +during training. However, with the assumption of the data +topology, it is possible to add some properties to the architecture +to reduce the number of parameters to learn and improve the +network implementation efficiency. These key ideas are local +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 4 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + +Algorithm +Back-propagation algorithm for a L-layer network with +weights (l) and a training set [(x1, yi), .., (xm, Ym)). +1 for l ← l to L do +2 +() = small random value ; // Initialise network weights for +each layer +3 end +4 foreach epoch do +5 +for l ← l to L do +△() = 0 ; +9 +// Initialise gradient matrices +7 +end +// For each training example +8 +foreach (Xi, yi) E [(x1, y1), . . . , (Xm, ym) do +// Forward propagation +w(1) ← Xi; +9 +10 +for l ← 2 to L do +w() ← g((1-1)w(-1) ; +11 +// For each layer of the +network +12 +end +// Back-propagation +s(L) ←w(L) -yi ; +13 +// Compute the error at the output +layer +14 +for l ← L - 1 to 2 do +15 +s() <← ((0())Ts()). * w(). * (1 - w() ; +// Compute the +error of each unit at the hidden layers +() ←() +s()(w()T ;// Update the matrix △ for +16 +each layer +17 +end +18 +end +// Gradient-descent: Update weights using learning rate +n and gradient +19 +for l ← l to L do +20 +()() () +21 +end +22 end +23 return 9(1).. +. 0(L)connections, shared weights, pooling, and the use of many layers +[32]. +The CNN units are arranged in three dimensions in each layer +of the network: width, height, and depth of the activation +volume. As depicted in Figure 2, a total of 3 different types of +layers are usually stacked to form the full CNN architecture: +convolutional layer, pooling layer, and fully connected layer. +Fully connected layers are layers of a traditional MLP, as +described in the section Feedforward Neural Networks. +Figure 2. Convolutional neural network architecture. +Convolutional Layer +The convolutional layer is the core layer of a CNN. The basic +idea is that instead of connecting a unit to every unit of the +previous layer, it is only connected to a local region of the +previous layer. The spatial extent of this connection is called +the receptive field of the unit or filter size. This is a +hyperparameter of the model. The filter size along the depth +axis is the same as that of the previous layer. This shows an +asymmetry in the way spatial dimensions (width and height) +and the depth dimension are treated, making the network +particularly adapted for multispectral images. The connectivity +of the convolutional layer is local along the width and height, +but the layer is fully connected along with depth. +A convolutional layer’s parameters can also be seen as a set of +spatially small-sized learnable filters or kernels. During the +forward pass, the filters are convolved across the width and +height dimensions of the input volume. This action produces a +2D activation map outputting the responses of the filter at each +position of the input layer [26,32]. The output volume of a +convolutional layer depends on three hyperparameters: the +number of filters, the stride, and zero padding. +The number of filters in the same receptive field determines the +depth of the output volume. A different filter activates for every +different pattern. A set of units with the same receptive field is +called the breadth of the output layer. +The stride used when the filters are slid along the spatial +dimensions of the previous layer affects the height and width +of the output volume. The higher the stride, the smaller is the +output volume. +The input volume can be padded with zeros around the border +to keep the information at the border. Without zero padding, +the information carried by the pixels at the border of the input +image vanishes quickly after successive convolutional layers. +This artificially increases the size of the input layer, thereby +increasing the size of the output layer. +Furthermore, the parameter-sharing scheme is used to reduce +the number of parameters to be learned. It is based on the +assumption that a useful feature at one position of the input +layer is also useful at a different position. This means that the +units on the same output depth slice use the same weights and +biases. This explains the fact that the forward propagation +through a convolutional layer is equivalent to convoluting a +filter or kernel with the input layer. +Pooling Layer +Typically, a pooling layer is inserted between the successive +convolution layers. The pooling function replaces the output of +a convolutional layer at a certain unit with the statistic of its +neighboring units. The most popular pooling function used is +the max-pooling method introduced by Zhou et al [33]. The +pooling layer aims to make the system invariant to small input +translations. This property gives more importance to whether +a feature is present in the input rather than its exact position. +CNN Feature Extraction and Classification +The combination of convolutional and pooling layers aims to +learn the best features that can be extracted from the data set. +This contrasts with most current methods that use handcrafted +feature extraction techniques, such as those presented in the +previous sections. These approaches can yield very good results +but are usually sensitive to the data set and perform poorly when +applied to different data sets. The combination of convolutional +and pooling layers of a CNN provides a more versatile method +for extracting features from images. The fully connected layers +of the CNN correspond to the classifier. It aims at learning to +classify learned features. As a result, a CNN is a unified versatile +scheme for feature extraction and classification. As medical +image classification is often a very complex task, it requires +carefully manufactured feature sets for each type of data or even +each different data set; doing just that with a unified framework, +CNNs seem particularly adapted to the field. +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 5 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + +Input layer +Comvolution layer +Pooling layer +Corvolution layer +Poolnglayer +Outputlayer +W, +Fully connected +ConvolutionandPoolingLayers +layersData Set Description +The prostate gland and the colorectum have a similar tissue +structure, with the tubular glandular mucosa—composed of +epithelium and lamina propria—being their main functional +tissue. This characteristic implies that these tissues are subject +to development of the same types of tumors and cancers. +Carcinomas are the most common type of malignant tumor and +they are derived from epithelial cells [34]. Carcinomas are called +adenocarcinomas when derived from glandular tissues, which +is the case for both organs studied in this paper. All growths +are not necessarily malignant, and benign polyps can occur [35]. +They are usually noncancerous growths of the mucosa into the +lumen and can be of different types. +Although most polyps are completely benign, such as +hyperplastic polyps or hyperplasia, some types of polyps can +transform into adenocarcinoma and can be considered as a +precancerous stage. They are called adenomas and can be tubular +or villous, depending on their growth patterns [36]. Hyperplastic +polyps are characterized by an increase in the number of cells, +resulting in an increased size of the tissue because of enhanced +cell division. In contrast to an adenoma or a carcinoma, the +division rate in a hyperplastic polyp returns to normal as soon +as the stimulus is removed. +To best describe the different types of tumor recognized by +pathologists, the following two data sets were used for the +purpose of this study: +1. +The prostate data set, which was used in previous works +by Tahir and Bouridane [13] and Peyret et al [17], consists +of 512 different multispectral prostate tumor tissue images +of size 128×128. The images were taken at 16 spectral +channels (500-650 nm) and 40× magnification power. The +samples were evaluated by 2 highly experienced +independent pathologists and labeled into four classes: 128 +cases of stroma, which is normal muscular tissue, 128 cases +of benign prostatic hyperplasia, a benign condition, 128 +cases of prostatic intraepithelial neoplasia, a precancerous +stage, and 128 cases of prostatic carcinoma, an abnormal +tissue development corresponding to cancer. +2. +The colorectal data set, which consists of multispectral +colorectal histology data with a 40× magnification power, +was developed by the University of Qatar in collaboration +with Al-Ahli Hospital, Doha. It splits into 4 classes, each +composed of 40 images. The images were acquired on a +wider spectrum than the first data set, as it was spread on +the visible and infrared ranges of the electromagnetic +spectrum with an interval of 23 nm between each +wavelength. That is to say, in the visible range, the +wavelength interval is 23 nm starting from 465 to 695 nm, +and in the infrared range, the wavelength interval is also +23 nm and ranges from 900 to 1590 nm. The special size +was 128×60 pixels. The 4 classes were defined as +carcinoma, containing images of cancerous colon biopsies; +tubular adenoma, a precancerous stage; hyperplastic polyp, +a benign polyp; and no remarkable pathology. +Experiments +Hardware and Software Specifications +To train deep CNNs, a GPU is required. The system used for +this experiment was equipped with 1 NVIDIA K80 GPU and +4 central processing units. It had 61-GB RAM. Regarding +software, Keras with a TensorFlow backend was used. Keras +has the advantage of making available deep learning models +alongside pretrained weights. +Selected Architecture +The proposed CNN architecture evaluated for the task at hand +was based on Visual Geometry Group 16 (VGG16) [37]. To +design the proposed architecture, the last block of the +convolutional layers of VGG16 was removed, and the number +of filters per layer was halved. The idea is to reduce the capacity +of the network because the interclass similarity in the data sets +used for the task was high compared with the data set on which +VGG16 was tested. +As represented in Figures 3 and 4, the overall proposed network +architecture consists of a total of 13 layers with weights—the +first 10 being convolutional layers, and the remaining 3 fully +connected layers. The output of the last fully connected layer +was fed to a SoftMax classifier, which is a generalization of the +logistic regression classifier to the multiclass problem and +produces a distribution of the 4 class labels. The network uses +cross-entropy as a loss function. +Similar to VGG16, we decided to use a small kernel with a size +of 3 pixels for every convolutional layer. The strategy of +stacking convolutional layers with a small filter size is preferred +to using a single large receptive eld convolutional layer. For +the same final receptive field, the former strategy includes +nonlinearities (ReLU functions) at each layer, whereas the latter +computes a simple linear function on the input, which makes +the features less expressive. A stride of 1 was also adopted for +the entire network to minimize information loss. +To achieve better control over the output size of each layer and +maintain border information, a zero padding of 1 is added before +each convolutional layer. The first 2 convolutional layers use +32 kernels followed by a 2 2 max-pooling layer. The +max-pooling layer reduces the size of the output and thus the +network capacity. The number of kernels is doubled in the next +convolutional layer to compensate for this loss. Consequently, +this sequence is followed by 2 convolutional layers with 64 +filters, and then a new max-pooling layer is applied. This is +followed by a series of 3 convolutional layers with 128 filters +and a max-pooling layer. A final series of 3 convolutional layers +with 256 filters and a max-pooling layer was applied. The +neurons in the 3 fully connected layers with sizes of 1024, 1024, +and 4, respectively, are connected to all neurons in the previous +layer. The ReLU nonlinearity was applied to the output of every +layer with weights. +Dropout is used after every max-pooling and fully connected +layer to reduce overfitting. An early stopping strategy is also +adopted to reduce the training time and regularization. Finally, +data augmentation is carried out using the following +transformations: each image is flipped along the 2 special axes, +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 6 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + +and 30 rotations in both directions are applied. This results in +the generation of 27 fake images for each real data image. To +ensure that the generalization is not overestimated, data set +augmentation is performed after splitting the data set into +training and test sets. +Figure 3. Illustration of the architecture of the proposed convolutional neural network for prostate cancer images. ReLU: rectified linear unit. +Figure 4. Illustration of the architecture of the proposed convolutional neural network for colorectal cancer images. ReLU: rectified linear unit. +Details of Learning +The weights of each layer are initialized using the Xavier +initialization method [38], where the weights are drawn from a +normal distribution centered on zero and with an SD of the +following: +where Nin and Nout are the numbers of input and output units, +respectively. The network was trained separately on the 2 data +sets. +The learning rate used was the same for all layers. It is optimized +using a grid-search scheme, the results of which are presented +in Figures 5 and 6. The learning rate selected for training was +0.0001 for both data sets. +For each model training, a 10-fold cross-validation technique +was adopted to obtain a good estimate of the systems’ +generalization accuracy. This provides a large training set for +better learning. +Figures 7 and 8 illustrate the evolution of the loss function +during training for the prostate and colorectal data sets, +respectively. Figures 9 and 10 show the evolution of their +accuracies. It can be observed from these figures that the +validation accuracy is very close to the training accuracy, which +proves that the model is not in the overfitting regime. The higher +variation in validation accuracy and loss can be explained by +the smaller set used for validation compared with that used for +training. +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 7 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + +256 x 256 x 16 +256 x 256 x 32 +127 ×127 ×64 +63x63x128 +15x15×256 +1 x 31 x 256 +1 × 1 × 1024 1 × 1 × 4 + convolution+ReLU +max pooling +fully connected+ReLU +xeujos128 x 160 x 42 +128 x 160 x 32 +63 × 79 × 64 +31x 39 x 128 +7x9x256 +5 x 19 x 256 +1 × 1 × 1024 1 × 1 × 4 + convolution+ReLU +max pooling +fully connected+ReLU +softmax2 +N Nin+NoutFigure 5. Validation accuracy obtained with different learning rates for the network trained on prostate data. +Figure 6. Validation accuracy obtained with different learning rates for the network trained on colorectal data. +Figure 7. Loss function evolution during training for the prostate data set. +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 8 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + +100 +90 +80 +70 +60 +50 +40 +30 - +TTTT +TTT +LL +10~7 +10~6 +10~s +10~4 +10~3 +TTTTTT +10~2 +10~1 +learning rate100 +90 +80 +ccuracy +70 +60 - +50 +40 +OE +10~7 +TTTTT +10~6 +TTTTTT +10~5 +TTTT +10~4 +TTTT +10~3 +TTTTT +10~2 +learning rateModel loss +1.75 +train +validation +1.50 +125 +0.75 +0.50 +0.25 +0.00 +0 +10 +20 +30 +40 +50 +epochFigure 8. Loss function evolution during training for the colorectal data set. +Figure 9. Accuracy evolution during training for the prostate data set. +Figure 10. Accuracy evolution during training for the colorectal data set. +Transfer Learning +Transfer learning consists of using a network previously trained +on another data set to use the knowledge acquired during this +learning task for the new task at hand [39]. In most transfer +learning for image classification tasks, the ImageNet data set +[40], which contains 1.2 million images with 1000 categories, +is used for pretraining the network. When only a small data set +is available, this allows the CNN to be trained on a very large +data set and therefore train a high-capacity network that captures +details without overfitting. Very deep networks also require a +lot of time and very powerful machines equipped with multiple +GPUs. Using pretrained networks can be advantageous when +appropriate resources are not provided. Several transfer-learning +scenarios are practical. +In the first scenario, the pretrained CNN is used as a fixed +feature extractor. The convolutional layers of the network are +kept with the weights determined during training on the +ImageNet data set, and the pretrained fully connected layers are +replaced with fully connected layers initialized with random +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 9 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + +Model loss +1.2 +train +1.0 +validation +0.8 +0.4 +0.2 +A +0.0 +0 +10 +20 +30 +40 +50 +60 +70 +EpochModel accuracy +1.0 - +0.8 + 0.6 +0.2 +train +0.0 +validation +0 +10 +20 +30 +40 +50 +epochModel accuracy +10 +0.9 +0.8 +0.6 +0.5 +train +0.4 +validation +0 +10 +20 +30 +40 +50 +60 +70 +Epochweights. During training, only the newly added fully connected +layers were marked as trainable. They used the features extracted +by pretrained convolutional layers as inputs. These features are +usually referred to as CNN codes [26,39]. +Another strategy is to retrain the fully connected layers from +scratch to fine-tune the weights of the pretrained convolutional +layers by continuing back-propagation. Either all the +convolutional layers can be retuned or only some of the +higher-level layers to avoid overfitting. This derives from the +observation that the lower-level layers usually learn more +generic features, such as edge detectors, that can be used for +many different learning tasks. In contrast, the high-level layers +tend to learn features that are more specific to the characteristics +of the classes of the original data set. +In this study, only the first scenario was investigated. The +pretrained CNNs are very deep and require very high +computational power to be retuned. Using them as feature +extractors is, in fact, equivalent to training only a relatively +shallow MLP. The proposed architecture was compared with +popular CNN architectures: VGG16 [37], InceptionV3 [21], +and ResNet50 [22]. These networks were initialized with the +weights obtained when pretraining them on the ImageNet data +set. However, InceptionV3 and ResNet50 are very deep +networks (48 and 152 layers, respectively), and a minimum +input image size is required. InceptionV3 requires a minimum +width and height of 139 pixels and ResNet50 of 197 pixels. The +images of the colorectal data set were smaller, and zero padding +was added to reach the required dimensions. Moreover, the +ImageNet images are RGB images and therefore have a depth +of 3 channels. To meet the dimension requirements, a principal +component analysis (PCA) was carried out to reduce the +dimensionality of the multiscale images to 3 channels. +Results and Discussion +Principal Results and Findings +To visualize the effect of the kernels on images through the +network, Figures 11 and 12 present examples of outputs of the +first convolutional layer of the networks trained with the prostate +and colorectal data sets, respectively. Figures 13 and 14 depict +examples of outputs of the last convolutional layers of the same +networks. It can be observed that after the first layer, the outputs +are very similar to the input image, for instance, with +transformations resembling edge detections. Once the image +has its own through the network, different regions or features +of the input image are represented in the outputs of the last +convolutional layer. Thus, the different layers learn a succession +of transformations, leading to an isolation of relevant regions +or features of the input image. The fully connected layers of the +network are then able to classify these particular features into +the 4 classes. +Figure 11. Example of an output of the first convolutional layer for the network trained on the prostate data set. +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 10 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + +U +100 +00 +00 +00 +00 +200 +00 +00 +00 +D +100200 +100200 +100200 +100200 +100200 +100 +00 +00 +200 +00 +00 +00 +00 +0 +100200 +0 +100200 +U +100200 +U +100200 +100200 +100200 +100 +00 +00 +00 +10 +200 +00 +20 +00 +00 +0 +100 +200 +100200 +100 +200 +100 +200 +100 +200 +100 +200 +100 +00 +00 +00 +bo +200 +00 +00 +00 +0 +100200 +100200 +100200 +0 +100200 +100200 +100200 +100 +00 +00 +00 +00 +0 +200 +200 +00 +00 +00 +0 +0 +200 +200 +100 +200 +100 +200 +0 +100 +200 +0 +100 +200 +100 +00 +200 +00 +100200 +100 +200Figure 12. Example of an output of the first convolutional layer for the network trained on the colorectal data set. +Figure 13. Example of an output of the last convolutional layer for the network trained on the prostate data set. +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 11 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + +0 + 50 +n +50 +50 +50 +100 +00 +1bo +0 +100 +0 +100 +100 +0 +100 +0 +100 +0 +100 +50 +50 +50 +100 +ibo +00 +lbo +Do +10 +0 +100 +0 +100 +100 +0 +100 +100 +0 +100 +0 +50 +50 +50 +50 +150 +50 +100 +00 +00 +0 +0 +100 +100 +100 +0 +100 +100 +0 +100 +0 + 50 +50 +100 +0 +100 +0 +100 +0 +100 +0 +100 +100 +0 +100 +0 +50 +50 +50 +50 +100 +DO +ibo +10 +100 +0 +100 +0 +100 +0 +100 +0 +100 +0 +100 +0 +50 +50 +100 +00 +0 +100 +0 +1000 +50 +25 +0 +25 +25 +25 +25 +25 +0 +25 +0 +25 +0 +25 +0 +25 +0 +25 +0 +25 +25 +25 +25 +0 +25 +0 +25 0 +25 +0 +25 +0 +25 +0 +25 0 +25 0 +25 +0 +25 +0 +25 +0 +25 +0 +25 +0 +25 0 +25 +0 +25 +0 +25Figure 14. Example of an output of the last convolutional layer for the network trained on the colorectal data set. +Table 1 displays the validation and test accuracies obtained +using the prostate and colorectal data sets for different CNN +models. This shows that the validation and test accuracies are +very close, proving a good generalization of the systems and +that overfitting was avoided. +The proposed CNN model achieved an average test accuracy +of 99.8% and 99.5% for the prostate and colorectal data sets, +respectively. Table 2 shows that the optimal CNN weights were +obtained after 44 and 70 epochs, respectively. The VGG16 +model initialized with Xavier weights trains very quickly for +the prostate data set; the optimal validation accuracy was +obtained after 19 epochs, as illustrated in Table 2. However, it +is less efficient at learning for the colorectal data set and requires +as many as 70 epochs to obtain the minimum validation loss. +The results also show slight overfitting for the colorectal data +set, as the validation accuracy is lower than the training +accuracy. This is because of the high capacity of the network. +When using this network with pretrained weights from +ImageNet, the training loss reaches a minimum after only a few +epochs, but the validation loss shows that the network overfits +marginally for both data sets. The test accuracy was also lower +than that of the proposed CNN by 99.5% and 98.1%, +respectively. This is because the CNN codes learned with the +ImageNet data set are not as adapted to the classification task +at hand as those learned with the proposed CNN. The +InceptionV3 model shows a higher overfitting and a lower +generalization for both data sets with 99.0% and 94.5% accuracy +for the prostate and colorectal data sets, respectively. This shows +once again that the CNN codes learned on the ImageNet data +set with this network are not adapted to the classification task +at hand. Finally, the pretrained ResNet50 achieved optimal +accuracy with the lowest number of epochs: 5 and 22 for the +prostate and colorectal data sets, respectively. It also achieves +100% average accuracy for the prostate data set, outperforming +the proposed CNN, and 99% for the colorectal data set, which +is slightly lower than the proposed data set. This lower +performance compared with the proposed CNN architecture for +the colorectal data set might be owing to some loss of +information when performing PCA on the 42 channels of the +colorectal data set images. The prostate data set consisted of +images with only 16 channels, and it is logical that the loss of +information is not as important during this transformation. +Therefore, the proposed CNN architecture is more adapted to +the task at hand than the other methods it was compared with. +However, ResNet50 shows very good performance when used +as a feature extractor and is trained with fewer epochs. In every +case, it was noted that the colorectal data set is more prone to +overfitting. This is probably owing to the size of the images, +which are spatially smaller than those for the prostate data set. +Therefore, a model with the correct capacity for the prostate +data set might be overestimated for the colorectal data set. +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 12 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + +0 +10 +010 +10 +0 +10 +010 +10 +010 +10 +010 +10.4 +010. +010.0 +010 +10.0 +2 +10 +10.0 +10.0 +010 +10 +10 +1 +0 +10 +10 +10 +10 +10 +10 +10 +010.0 +10 +10 +010.. +10 +10 +10.0 +100 +10.0 +10 +10 +0 +10 +10 +10. +0.10 +010g +10 +0 +010. +10 +10 +0 +0.10 +10 +110 +10 +0 +10 +10 +01000100 +10 +0 +010g +010 +010.。 +10 +10 +10 +10 +010。 +0.10.0 +0.1000 +100 +0100010. +0.10.0 +10.c +10 +10 +0 +10 +0 +0 +10 +0 10 +0 10 +0 +10 +0 10 +0 10 +0 10 +0 10 +0 10 +0 10 +0 +10 +0 10 +0 +10 +0 10Table 1. Validation and test accuracy comparison of different architectures.a +Colorectal data set (% accuracy), mean (SD) +Prostate data set (% accuracy), mean (SD) +Method +Test +Validation +Test +Validation +99.5 (0.1) +100 +99.8 (0.1) +100 +Proposed CNNb +99.2 (0.1) +99.0 (0.1) +99.6 (0.1) +100 +VGG16c Xavier initial +98.1 (0.1) +97.5 (0.2) +99.5 (0.1) +100 +VGG16 pretrained +94.5 (0.3) +92.3 (0.3) +99.0 (0.1) +98.8 (0.2) +InceptionV3 pretrain +99.0 (0.2) +99.5 (0.1) +100 +100 +ResNet50 pretrained +aSD values have been provided wherever applicable. +bCNN: convolutional neural network. +cVGG16: Visual Geometry Group 16. +Table 2. Number of epochs until early stopping. +Colorectal data set +Prostate data set +Method +70 +44 +Proposed CNNa +70 +19 +VGG16b Xavier initialization +38 +10 +VGG16 pretrained +53 +48 +InceptionV3 pretrained +22 +5 +ResNet50 pretrained +aCNN: convolutional neural network. +bVGG16: Visual Geometry Group 16. +Comparison Against Other Machine Learning Methods +Table 3 shows the test accuracy of the best-performing CNN +architectures compared with other methods from Tahir et al +[15], Bouatemane et al [16], Haj-Hassan et al [25], and Peyret +et al [17] stacked multispectral multiscale local binary pattern +(MMLBP) + gray-level co-occurrence matrix (GLCM), and +concatenated local binary pattern [18]. Regarding the prostate +data set, 5 systems have an accuracy above 99%: Bouatemane +et al [16], Stacked MMLBP+GLCM, the proposed CNN, +Haj-Hassan et al [25], and ResNet50 with pretrained weights. +The highest classification accuracy was achieved using +ResNet50 with 100% accuracy. The proposed CNN and the +study by Bouatemane et al [16] achieved 99.8% accuracy; +however, the SD was not given for the latter. Therefore, it is +not possible to determine the precision of the accuracy +estimation. The stacked MMLBP+GLCM system achieves +99.5% (SD 0.3 pp), which makes this performance similar to +that of the proposed CNN. However, a higher SD shows lower +precision in the accuracy estimation. Therefore, the proposed +CNN was preferred. The study by Haj-Hassan et al [25] achieved +a 99.17% accuracy with segmentation. Their system without +this preprocessing phase achieved an accuracy of 79.23%. +This can be explained by the lower capacity of their model +compared with ours. This has the advantage of requiring less +processing power. However, this is counterbalanced by the fact +that their system requires a preprocessing phase with the +intervention of a pathologist, which dramatically increases the +processing time of the system. Furthermore, they state that their +CNN model requires 500 epochs to be trained, which is much +higher than that of the proposed model. With respect to the +colorectal data set. Peyret et al [17] stacked MMLBP+GLCM +system and the proposed CNN both provided the same accuracy +and SD. They outperform ResNet50 with pretrained weights +by 0.5 pp. +Finally, when considering the results obtained with both data +sets, the stacked MMLBP+GLCM system and the proposed +CNN appear to provide the most stable results as well as the +highest accuracy. However, on average, the SD of the accuracy +achieved by the proposed CNN is lower than that obtained with +the stacked MMLBP+GLCM system. The performance of the +ResNet50 network seems to be more dependent on the data set +used. Moreover, it would be interesting to compare the system +proposed by Bouatemane et al [16] using the colorectal data set +to verify whether it performs as well on different data sets. +Considering the current information available on the system +performance and with the data sets available, the proposed CNN +is selected as the best-performing system in terms of accuracy +for the classification task at hand. +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 13 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + +Table 3. Accuracy comparison against other methods.a +Colorectal data set (% accuracy), mean (SD) +Prostate data set (% accuracy), mean (SD) +Method +Tahir et al [15] +• +• +N/Ab +98.9 +Bouatemane et al [16] +• +• +N/A +99.83 +Concatenated LBPc [18] +• +• +88.2 (0.5) +92.4 (0.4) +• +• +99.5 (0.3) +99.5 (0.1) +Stacked MMLBPd + GLCMe [41] +• +• +99.5 (0.1) +99.5 (0.3) +Haj-Hassan et al [25] +• +• +N/A +99.17 +Proposed CNN +• +• +99.5 (0.1) +99.8 (0.1) +ResNet50 pretrained +• +• +99.0 (0.2) +100 +aSD values have been provided wherever applicable. +bN/A: not applicable. +cLBP: local binary pattern. +dMMLBP: multispectral multiscale local binary pattern. +eGLCM: gray-level co-occurrence matrix. +Computational Complexity Analysis +In computer-aided diagnosis systems (CADSs), an unlabeled +image is fed to a previously trained system. Consequently, the +time used to process this image is decisive, as it is crucial that +the CADS works on the web. However, the forward pass of an +image through the CNN architectures studied in this study is +computationally +nonexpensive. +Table +4 +displays +the +classification times per image for all CNN architectures tested. +This demonstrates that only a few milliseconds are required to +classify one image once the CNN has been trained. However, +it must be noted that the proposed CNN architecture is much +quicker at classifying images than the others. This is because, +for the architectures described in the literature and the pretrained +networks, a PCA must be carried out to reduce to 3 the number +of channels of the image to be classified. This preprocessing +stage lengthens the total classification time. +As mentioned, training is performed only once when a CADS +is created. Consequently, training time is not a critical measure +of the problem at hand. However, the computational complexity +of deep learning systems can rapidly increase significantly. +Such architectures require high-performing hardware, including +GPUs. Some extremely deep architectures can also entail several +weeks of training time [26]. Such long training times +considerably slowed down the CADS development process. To +verify that the proposed system can be trained within a +reasonable duration, a comparison of the training times for each +architecture was carried out (Table 5). The computational times +depending on the hardware and software used, it is not possible +to compare the CNN architectures with other classification +systems proposed in other published works. However, this is +one of the first attempts to use deep learning for this application. +Therefore, this section aims to establish the ability of deep +learning systems to be trained in a short period using the data +sets used. +Unsurprisingly, Table 5 demonstrates that pretrained networks +have a much shorter training time per epoch owing to the +reduced number of layers to be trained; ResNet50 and +InceptionV3 can be trained in a few minutes. When considering +this measure of performance, the best architecture was +ResNet50. However, the total training time for every CNN +model is <2 hours, making it a reasonable time for developing +a CADS. +Table 4. Average convolutional neural network (CNN) classification computation times for 1 image. +Colorectal data set (ms) +Prostate data set (ms) +Method +7 +14 +Proposed CNN +42 +75 +VGG16a Xavier initial +42 +75 +VGG16 pretrained +42 +63 +InceptionV3 pretrained +47 +65 +ResNet50 pretrained +aVGG16: Visual Geometry Group 16. +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 14 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + +Table 5. Average convolutional neural network (CNN) training computation times for the complete data set. +Colorectal data set (seconds) +Prostate data set (seconds) +Method +Total training +Time per epoch +Total training +Time per epoch +2925 +45 +3780 +90 +Proposed CNN +6790 +97 +4655 +245 +VGG16a Xavier initial +1400 +35 +3154 +83 +VGG16 pretrained +705 +15 +1755 +39 +InceptionV3 pretrained +704 +32 +205 +41 +ResNet50 pretrained +aVGG16: Visual Geometry Group 16. +Conclusions +In this paper, the proposed CNN architecture was detailed and +compared with previously trained network models used as +feature extractors. These CNNs were also compared with other +classification methods from other published studies. The +proposed CNN demonstrated excellent performance compared +with pretrained CNNs and other classification methods. The +computational complexity of the CNNs was also analyzed, and +it was demonstrated that the proposed CNN is faster at +classifying images than pretrained networks because it avoids +a preprocessing phase. The conclusion of this overall analysis +is that the proposed CNN architecture was globally the +best-performing system for classifying colorectal and prostate +tumor images. +Acknowledgments +This research project was supported by a grant from the Research Supporting Program (Project Number: RSP2022R281), King +Saud University, Riyadh, Saudi Arabia. +Conflicts of Interest +None declared. +References +1. +Ferlay J, Soerjomataram I, Dikshit R, Eser S, Mathers C, Rebelo M, et al. Cancer incidence and mortality worldwide: +sources, methods and major patterns in GLOBOCAN 2012. Int J Cancer 2015 Mar 01;136(5):E359-E386 [FREE Full text] +[doi: 10.1002/ijc.29210] [Medline: 25220842] +2. +Heidenreich A, Bellmunt J, Bolla M, Joniau S, Mason M, Matveev V, European Association of Urology. 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Neurocomputing +2018 Jan;275(C):83-93 [FREE Full text] [doi: 10.1016/j.neucom.2017.05.010] +Abbreviations +CADS: computer-aided diagnosis system +CNN: convolutional neural network +GLCM: gray-level co-occurrence matrix +GPU: graphic processing unit +MLP: multilayer perceptron +MMLBP: multispectral multiscale local binary pattern +PCA: principal component analysis +ReLU: rectified linear unit +RGB: red, green, blue +VGG16: Visual Geometry Group 16 +Edited by A Mavragani; submitted 23.01.21; peer-reviewed by M Wu, SM Mir Hosseini; comments to author 25.06.21; revised version +received 08.09.21; accepted 11.12.21; published 09.02.22 +Please cite as: +Peyret R, alSaeed D, Khelifi F, Al-Ghreimil N, Al-Baity H, Bouridane A +Convolutional Neural Network–Based Automatic Classification of Colorectal and Prostate Tumor Biopsies Using Multispectral +Imagery: System Development Study +JMIR Bioinform Biotech 2022;3(1):e27394 +URL: https://bioinform.jmir.org/2022/1/e27394 +doi: 10.2196/27394 +PMID: +©Remy Peyret, Duaa alSaeed, Fouad Khelifi, Nadia Al-Ghreimil, Heyam Al-Baity, Ahmed Bouridane. Originally published in +JMIR Bioinformatics and Biotechnology (https://bioinform.jmir.org), 09.02.2022. This is an open-access article distributed under +the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted +use, distribution, and reproduction in any medium, provided the original work, first published in JMIR Bioinformatics and +Biotechnology, is properly cited. The complete bibliographic information, a link to the original publication on +https://bioinform.jmir.org/, as well as this copyright and license information must be included. +JMIR Bioinform Biotech 2022 | vol. 3 | iss. 1 | e27394 | p. 17 +https://bioinform.jmir.org/2022/1/e27394 +(page number not for citation purposes) +Peyret et al +JMIR BIOINFORMATICS AND BIOTECHNOLOGY +XSL•FO +RenderX + diff --git a/YNFPT4oBgHgl3EQftDXh/content/tmp_files/load_file.txt b/YNFPT4oBgHgl3EQftDXh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c280f25bb4bfe19af47c55204ea11e4d1dae1428 --- /dev/null +++ b/YNFPT4oBgHgl3EQftDXh/content/tmp_files/load_file.txt @@ -0,0 +1,1184 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf,len=1183 +page_content='Original Paper Convolutional Neural Network–Based Automatic Classification of Colorectal and Prostate Tumor Biopsies Using Multispectral Imagery: System Development Study Remy Peyret1, PhD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Duaa alSaeed2, PhD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Fouad Khelifi1, PhD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Nadia Al-Ghreimil2, PhD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Heyam Al-Baity2, PhD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Ahmed Bouridane1, PhD 1Northumbria University at Newcastle, Newcastle, United Kingdom 2College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia Corresponding Author: Duaa alSaeed, PhD College of Computer and Information Sciences King Saud University King Abdullah Road Riyadh, 11451 Saudi Arabia Phone: 966 555442477 Email: dalsaeed@ksu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='sa Abstract Background: Colorectal and prostate cancers are the most common types of cancer in men worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' To diagnose colorectal and prostate cancer, a pathologist performs a histological analysis on needle biopsy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This manual process is time-consuming and error-prone, resulting in high intra- and interobserver variability, which affects diagnosis reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Objective: This study aims to develop an automatic computerized system for diagnosing colorectal and prostate tumors by using images of biopsy samples to reduce time and diagnosis error rates associated with human analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Methods: In this study, we proposed a convolutional neural network (CNN) model for classifying colorectal and prostate tumors from multispectral images of biopsy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The key idea was to remove the last block of the convolutional layers and halve the number of filters per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Results: Our results showed excellent performance, with an average test accuracy of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='8% and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5% for the prostate and colorectal data sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The system showed excellent performance when compared with pretrained CNNs and other classification methods, as it avoids the preprocessing phase while using a single CNN model for the whole classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Overall, the proposed CNN architecture was globally the best-performing system for classifying colorectal and prostate tumor images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Conclusions: The proposed CNN architecture was detailed and compared with previously trained network models used as feature extractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' These CNNs were also compared with other classification techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' As opposed to pretrained CNNs and other classification approaches, the proposed CNN yielded excellent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The computational complexity of the CNNs was also investigated, and it was shown that the proposed CNN is better at classifying images than pretrained networks because it does not require preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Thus, the overall analysis was that the proposed CNN architecture was globally the best-performing system for classifying colorectal and prostate tumor images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' (JMIR Bioinform Biotech 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='3(1):e27394) doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='2196/27394 KEYWORDS convolutional neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' colorectal tumor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' prostate tumor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' machine learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' image processing JMIR Bioinform Biotech 2022 | vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 3 | iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 1 | e27394 | p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 1 https://bioinform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='jmir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='org/2022/1/e27394 (page number not for citation purposes) Peyret et al JMIR BIOINFORMATICS AND BIOTECHNOLOGY XSL•FO RenderX Introduction Background According to the World Health Organization 2014 report, 14 million new cases of cancer were diagnosed in 2012, and the disease caused 8 million people to die in the same period [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Colorectal cancer is the third most common cancer globally, whereas prostate cancer is the second most common cancer among men, accounting for 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='7% and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='9% of all cancers in both sexes, respectively [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Both colorectal and prostate tissues are glandular and therefore have a similar histological appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' For prostate cancer diagnosis, the European Association of Urology guidelines [2] recommend the performance of a histological analysis on a sample taken from a needle biopsy by a pathologist who decides the grade and stage of cancer or the type of tumor based on their experience and expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, this process is time consuming and it also results in a high intra- and interobserver variability [3,4], which affects diagnosis reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' In December 1999, a study [5] of more than 6000 patients conducted by Johns Hopkins researchers found that up to 2 out of every 100 people who came to larger medical centers for treatment were given an incorrect diagnosis after histological analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' These results suggest that second-opinion pathology examinations not only prevent errors but also save lives and money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Consequently, there is an increasing interest among pathology experts in the use of machine vision (or computational diagnosis tools) to reduce diagnosis error rates by lowering the fallible aspect of human image interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Computer-aided diagnosis can assist pathologists in reducing the human analysis time, improving efficiency, and acting as a second opinion [6-8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Adding computer-based quantitative analysis to human qualitative interpretation could significantly reduce the intra- and interobserver variability revealed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The main objective of this study is to develop an automatic computerized system for the diagnosis of colorectal and prostate tumors using images of biopsy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Numerous investigations concerning prostate or colorectal tumor classification have been carried out [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, most use color spaces limited to gray-scale or red, green, blue (RGB) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' In the last decade, many studies have used multispectral images [11-18], which are acquired using a more precise sampling of the light spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This approach aims to better capture the spectrum of the reflected light coming from the observed sample, offering more discriminative information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Lasch et al [19] suggested that multispectral imagery can improve histopathological analysis by capturing patterns that are invisible to the human vision system and standard RGB imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Multispectral imaging studies have shown promising results and often outperformed systems using traditional gray-scale or RGB images [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, multispectral images contain a large amount of data, making them more difficult to process because of increased execution time and problems caused by the curse of dimensionality [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Since the emergence of graphic processing units (GPUs) with sufficient processing power to train Convolutional neural networks (CNNs) in 2011, these models have seen a growing interest in image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Several models have been developed and tested on the ImageNet data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' As an example, the AlexNet architecture was developed in 2012 [20] and won several international competitions, including the ImageNet competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' GoogLeNet [21], a 22 layers deep network, won the ImageNet competition of 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' He et al [22] deepened the networks even more with ResNet and won the best paper in 2015 at the Conference on Computer Vision and Pattern Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' To reduce training times, they developed a framework in which layers are formulated as a residual function with reference to the layer input, as opposed to the unreferenced learning functions previously used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The residual network comprised 152 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' In 2016, Google DeepMind used a mix of supervised deep learning and reinforcement learning (ie, deep reinforcement learning) to create a system capable of learning how to play the game of Go [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This program, called AlphaGo, achieved a 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='8% winning rate against other Go programs and defeated the human European Go champion by 5 games to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' In 2017, they created AlphaGo Zero [24], which outperformed the original AlphaGo in terms of performance and learning time without using any human knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' CNNs seem particularly adapted to the problem of microscopic images of tumor classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' A previous study [25] applied CNNs to microscopic images of colorectal cancer and found a promising accuracy of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, in this study, images were preprocessed using an active contour model before being fed to the CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This operation requires the intervention of a pathologist to select the region of interest from the segmented image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Otherwise, this step can be replaced by another supervised learning model, which requires more training and thus dramatically increases the processing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This study proposes a model that does not require a preprocessing phase and uses a single CNN model for the entire classification task using multispectral images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Deep learning is a branch of machine learning that attempts to mimic the thinking process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' To process data, information is passed through a network consisting of different layers, where each layer serves as input to the following layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The first layer of a network is referred to as the input layer, whereas the last layer is the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' All the layers in between are called hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Typically, a layer is a simple algorithm that consists of an activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This field of machine learning is now very active, and the research community is focused on solving practical applications using modern deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This study aims to apply the deep learning framework to the problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Objective The primary objective of this study is to develop a computerized automatic system for the diagnosis of colorectal and prostate tumors using images of biopsy samples to reduce time and diagnosis error rates associated with human analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' To achieve this, we propose a CNN model for the classification of colorectal and prostate tumors from multispectral images of biopsy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The key idea is based on removing the last block of JMIR Bioinform Biotech 2022 | vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 3 | iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 1 | e27394 | p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 2 https://bioinform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='jmir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='org/2022/1/e27394 (page number not for citation purposes) Peyret et al JMIR BIOINFORMATICS AND BIOTECHNOLOGY XSL•FO RenderX the convolutional layers and halving the number of filters per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This paper is organized as follows: we first describe the principles of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The second section discusses the proposed method, whereas the data sets of multispectral tumor images are described in the third section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' In the fourth section, the experiments carried out to validate the approach are detailed, and finally their results are presented and analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Feedforward Neural Networks Overview Feedforward neural networks, also called multilayer perceptrons (MLPs), are the basis of deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' They aim to approximate the function f:~x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='y, where ~x is an input feature vector and y is its corresponding class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The network builds a mapping ~y=f(~x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=') by learning the parameters that provide the best approximation function to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' In this type of network, information moves from the input to the output through intermediate layers with no feedback connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The number of layers is called the network depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Each layer consists of a vector of functions or units that act in parallel, and the dimension of this vector is the width of the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Therefore, many hyperparameters need to be chosen when designing a neural network model, including its architecture, that is, the number of layers and units per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' A hidden layer computes an affine transformation of its input and then applies a nonlinear function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This is defined by h=g(W~x+b), where h is the output of the hidden layer, W is the weight of the affine transformation, and b is the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' W and b are the parameters learned when training the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The function chosen for each unit is called the activation function and is inspired by the behavior of biological neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The most widely used activation function is the rectified linear unit (ReLU), defined by g(z)=max(0, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Many other options are available, and the research on activation function is still a very active field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, the ReLU has proven to perform well and is the default choice for activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Network training is performed using gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The main difference from other models is that the nonlinearity of neural networks causes the loss function to be nonconvex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Unlike convex optimization used with support vector machines or deep reinforcement learning, there is no guarantee of global convergence of a gradient descent applied to a nonconvex loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Consequently, the learning process is sensitive to the initial values of weights and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' To apply gradient-based learning, a cost function must be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The problem at hand in this study defines a conditional distribution p(y|x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' θ) and the maximum likelihood principle is well adapted for it [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' As a result, the cross-entropy between the training data and the model’s prediction, which is equivalent to the negative log-likelihood, is used as the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' It enables the model to estimate the conditional probability of the classes if the input is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The cost function model is as follows: where is the distribution of the training data and pmodel is the model distribution and the set of parameters for which the cost function is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Consequently, the specific form of the cost function changes depending on the form of the log pmodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Back-Propagation During training, the gradient of the cost function ΔθJ (θ) is computed using a back-propagation algorithm [27-29] to allow information to flow backward through the network and compute the error made on each network weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' A gradient descent was then used to minimize the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Learning was subsequently performed by updating the weights of the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This procedure is described in the algorithm shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Training a neural network consists of applying a series of forwarding propagations—the network output is generated from the data through the network, and back-propagations compute the error at each unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Each of these forward propagation and back-propagation combinations is called a pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' A pass of all the training examples is performed to compute the gradient used for the gradient-descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' A pass of every training example is called an epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' At the end of each epoch, the network weights are updated using a learning rate hyperparameter, which is multiplied by the gradient calculated with back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The learning rate is one of the most important hyperparameters for tuning in a neural network, as it controls the effective capacity of the network [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Therefore, it needs to be carefully optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' If the learning rate is too large, the gradient descent can have the opposite of the desired effect, and training accuracy can decrease [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, when it is too small, the training is slower, and sometimes the training accuracy can stay permanently small [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The number of epochs is also a hyperparameter that can be tuned ahead of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' JMIR Bioinform Biotech 2022 | vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 3 | iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 1 | e27394 | p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 3 https://bioinform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='jmir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='org/2022/1/e27394 (page number not for citation purposes) Peyret et al JMIR BIOINFORMATICS AND BIOTECHNOLOGY XSL•FO RenderX J(O) = -Ex,Y~pdata log Pmodel(y|x)PdataFigure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Back-propagation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Methods Overview As previously mentioned, the research community is now focusing on solving practical applications using deep learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Our proposed solution to the problem of diagnosing colorectal and prostate cancer is to apply a deep learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' CNNs [27,31] are a type of neural network that specialize in data with a grid-like topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' They are particularly adapted for image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Similar to conventional neural networks, they consist of units with weights and biases that are learned during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, with the assumption of the data topology, it is possible to add some properties to the architecture to reduce the number of parameters to learn and improve the network implementation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' These key ideas are local JMIR Bioinform Biotech 2022 | vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 3 | iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 1 | e27394 | p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 4 https://bioinform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='jmir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='org/2022/1/e27394 (page number not for citation purposes) Peyret et al JMIR BIOINFORMATICS AND BIOTECHNOLOGY XSL•FO RenderX Algorithm Back-propagation algorithm for a L-layer network with weights (l) and a training set [(x1, yi), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='., (xm, Ym)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 1 for l ← l to L do 2 () = small random value ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' // Initialise network weights for each layer 3 end 4 foreach epoch do 5 for l ← l to L do △() = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 9 // Initialise gradient matrices 7 end // For each training example 8 foreach (Xi, yi) E [(x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' , (Xm, ym) do // Forward propagation w(1) ← Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 9 10 for l ← 2 to L do w() ← g((1-1)w(-1) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 11 // For each layer of the network 12 end // Back-propagation s(L) ←w(L) -yi ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 13 // Compute the error at the output layer 14 for l ← L - 1 to 2 do 15 s() <← ((0())Ts()).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' * w().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' * (1 - w() ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' // Compute the error of each unit at the hidden layers () ←() +s()(w()T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='// Update the matrix △ for 16 each layer 17 end 18 end // Gradient-descent: Update weights using learning rate n and gradient 19 for l ← l to L do 20 ()() () 21 end 22 end 23 return 9(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 0(L)connections, shared weights, pooling, and the use of many layers [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The CNN units are arranged in three dimensions in each layer of the network: width, height, and depth of the activation volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' As depicted in Figure 2, a total of 3 different types of layers are usually stacked to form the full CNN architecture: convolutional layer, pooling layer, and fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Fully connected layers are layers of a traditional MLP, as described in the section Feedforward Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Convolutional neural network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Convolutional Layer The convolutional layer is the core layer of a CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The basic idea is that instead of connecting a unit to every unit of the previous layer, it is only connected to a local region of the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The spatial extent of this connection is called the receptive field of the unit or filter size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This is a hyperparameter of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The filter size along the depth axis is the same as that of the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This shows an asymmetry in the way spatial dimensions (width and height) and the depth dimension are treated, making the network particularly adapted for multispectral images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The connectivity of the convolutional layer is local along the width and height, but the layer is fully connected along with depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' A convolutional layer’s parameters can also be seen as a set of spatially small-sized learnable filters or kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' During the forward pass, the filters are convolved across the width and height dimensions of the input volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This action produces a 2D activation map outputting the responses of the filter at each position of the input layer [26,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The output volume of a convolutional layer depends on three hyperparameters: the number of filters, the stride, and zero padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The number of filters in the same receptive field determines the depth of the output volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' A different filter activates for every different pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' A set of units with the same receptive field is called the breadth of the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The stride used when the filters are slid along the spatial dimensions of the previous layer affects the height and width of the output volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The higher the stride, the smaller is the output volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The input volume can be padded with zeros around the border to keep the information at the border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Without zero padding, the information carried by the pixels at the border of the input image vanishes quickly after successive convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This artificially increases the size of the input layer, thereby increasing the size of the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Furthermore, the parameter-sharing scheme is used to reduce the number of parameters to be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' It is based on the assumption that a useful feature at one position of the input layer is also useful at a different position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This means that the units on the same output depth slice use the same weights and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This explains the fact that the forward propagation through a convolutional layer is equivalent to convoluting a filter or kernel with the input layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Pooling Layer Typically, a pooling layer is inserted between the successive convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The pooling function replaces the output of a convolutional layer at a certain unit with the statistic of its neighboring units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The most popular pooling function used is the max-pooling method introduced by Zhou et al [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The pooling layer aims to make the system invariant to small input translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This property gives more importance to whether a feature is present in the input rather than its exact position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' CNN Feature Extraction and Classification The combination of convolutional and pooling layers aims to learn the best features that can be extracted from the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This contrasts with most current methods that use handcrafted feature extraction techniques, such as those presented in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' These approaches can yield very good results but are usually sensitive to the data set and perform poorly when applied to different data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The combination of convolutional and pooling layers of a CNN provides a more versatile method for extracting features from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The fully connected layers of the CNN correspond to the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' It aims at learning to classify learned features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' As a result, a CNN is a unified versatile scheme for feature extraction and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' As medical image classification is often a very complex task, it requires carefully manufactured feature sets for each type of data or even each different data set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' doing just that with a unified framework, CNNs seem particularly adapted to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' JMIR Bioinform Biotech 2022 | vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 3 | iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 1 | e27394 | p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 5 https://bioinform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='jmir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='org/2022/1/e27394 (page number not for citation purposes) Peyret et al JMIR BIOINFORMATICS AND BIOTECHNOLOGY XSL•FO RenderX Input layer Comvolution layer Pooling layer Corvolution layer Poolnglayer Outputlayer W, Fully connected ConvolutionandPoolingLayers layersData Set Description The prostate gland and the colorectum have a similar tissue structure, with the tubular glandular mucosa—composed of epithelium and lamina propria—being their main functional tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This characteristic implies that these tissues are subject to development of the same types of tumors and cancers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Carcinomas are the most common type of malignant tumor and they are derived from epithelial cells [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Carcinomas are called adenocarcinomas when derived from glandular tissues, which is the case for both organs studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' All growths are not necessarily malignant, and benign polyps can occur [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' They are usually noncancerous growths of the mucosa into the lumen and can be of different types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Although most polyps are completely benign, such as hyperplastic polyps or hyperplasia, some types of polyps can transform into adenocarcinoma and can be considered as a precancerous stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' They are called adenomas and can be tubular or villous, depending on their growth patterns [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Hyperplastic polyps are characterized by an increase in the number of cells, resulting in an increased size of the tissue because of enhanced cell division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' In contrast to an adenoma or a carcinoma, the division rate in a hyperplastic polyp returns to normal as soon as the stimulus is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' To best describe the different types of tumor recognized by pathologists, the following two data sets were used for the purpose of this study: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The prostate data set, which was used in previous works by Tahir and Bouridane [13] and Peyret et al [17], consists of 512 different multispectral prostate tumor tissue images of size 128×128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The images were taken at 16 spectral channels (500-650 nm) and 40× magnification power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The samples were evaluated by 2 highly experienced independent pathologists and labeled into four classes: 128 cases of stroma, which is normal muscular tissue, 128 cases of benign prostatic hyperplasia, a benign condition, 128 cases of prostatic intraepithelial neoplasia, a precancerous stage, and 128 cases of prostatic carcinoma, an abnormal tissue development corresponding to cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The colorectal data set, which consists of multispectral colorectal histology data with a 40× magnification power, was developed by the University of Qatar in collaboration with Al-Ahli Hospital, Doha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' It splits into 4 classes, each composed of 40 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The images were acquired on a wider spectrum than the first data set, as it was spread on the visible and infrared ranges of the electromagnetic spectrum with an interval of 23 nm between each wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' That is to say, in the visible range, the wavelength interval is 23 nm starting from 465 to 695 nm, and in the infrared range, the wavelength interval is also 23 nm and ranges from 900 to 1590 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The special size was 128×60 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The 4 classes were defined as carcinoma, containing images of cancerous colon biopsies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' tubular adenoma, a precancerous stage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' hyperplastic polyp, a benign polyp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' and no remarkable pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Experiments Hardware and Software Specifications To train deep CNNs, a GPU is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The system used for this experiment was equipped with 1 NVIDIA K80 GPU and 4 central processing units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' It had 61-GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Regarding software, Keras with a TensorFlow backend was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Keras has the advantage of making available deep learning models alongside pretrained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Selected Architecture The proposed CNN architecture evaluated for the task at hand was based on Visual Geometry Group 16 (VGG16) [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' To design the proposed architecture, the last block of the convolutional layers of VGG16 was removed, and the number of filters per layer was halved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The idea is to reduce the capacity of the network because the interclass similarity in the data sets used for the task was high compared with the data set on which VGG16 was tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' As represented in Figures 3 and 4, the overall proposed network architecture consists of a total of 13 layers with weights—the first 10 being convolutional layers, and the remaining 3 fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The output of the last fully connected layer was fed to a SoftMax classifier, which is a generalization of the logistic regression classifier to the multiclass problem and produces a distribution of the 4 class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The network uses cross-entropy as a loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Similar to VGG16, we decided to use a small kernel with a size of 3 pixels for every convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The strategy of stacking convolutional layers with a small filter size is preferred to using a single large receptive eld convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' For the same final receptive field, the former strategy includes nonlinearities (ReLU functions) at each layer, whereas the latter computes a simple linear function on the input, which makes the features less expressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' A stride of 1 was also adopted for the entire network to minimize information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' To achieve better control over the output size of each layer and maintain border information, a zero padding of 1 is added before each convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The first 2 convolutional layers use 32 kernels followed by a 2 2 max-pooling layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The max-pooling layer reduces the size of the output and thus the network capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The number of kernels is doubled in the next convolutional layer to compensate for this loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Consequently, this sequence is followed by 2 convolutional layers with 64 filters, and then a new max-pooling layer is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This is followed by a series of 3 convolutional layers with 128 filters and a max-pooling layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' A final series of 3 convolutional layers with 256 filters and a max-pooling layer was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The neurons in the 3 fully connected layers with sizes of 1024, 1024, and 4, respectively, are connected to all neurons in the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The ReLU nonlinearity was applied to the output of every layer with weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Dropout is used after every max-pooling and fully connected layer to reduce overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' An early stopping strategy is also adopted to reduce the training time and regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Finally, data augmentation is carried out using the following transformations: each image is flipped along the 2 special axes, JMIR Bioinform Biotech 2022 | vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 3 | iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 1 | e27394 | p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 6 https://bioinform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='jmir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='org/2022/1/e27394 (page number not for citation purposes) Peyret et al JMIR BIOINFORMATICS AND BIOTECHNOLOGY XSL•FO RenderX and 30 rotations in both directions are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This results in the generation of 27 fake images for each real data image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' To ensure that the generalization is not overestimated, data set augmentation is performed after splitting the data set into training and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Illustration of the architecture of the proposed convolutional neural network for prostate cancer images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' ReLU: rectified linear unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Illustration of the architecture of the proposed convolutional neural network for colorectal cancer images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' ReLU: rectified linear unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Details of Learning The weights of each layer are initialized using the Xavier initialization method [38], where the weights are drawn from a normal distribution centered on zero and with an SD of the following: where Nin and Nout are the numbers of input and output units, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The network was trained separately on the 2 data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The learning rate used was the same for all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' It is optimized using a grid-search scheme, the results of which are presented in Figures 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The learning rate selected for training was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='0001 for both data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' For each model training, a 10-fold cross-validation technique was adopted to obtain a good estimate of the systems’ generalization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This provides a large training set for better learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Figures 7 and 8 illustrate the evolution of the loss function during training for the prostate and colorectal data sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Figures 9 and 10 show the evolution of their accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' It can be observed from these figures that the validation accuracy is very close to the training accuracy, which proves that the model is not in the overfitting regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The higher variation in validation accuracy and loss can be explained by the smaller set used for validation compared with that used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' JMIR Bioinform Biotech 2022 | vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 3 | iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 1 | e27394 | p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 7 https://bioinform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='jmir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='org/2022/1/e27394 (page number not for citation purposes) Peyret et al JMIR BIOINFORMATICS AND BIOTECHNOLOGY XSL•FO RenderX 256 x 256 x 16 256 x 256 x 32 127 ×127 ×64 63x63x128 15x15×256 1 x 31 x 256 1 × 1 × 1024 1 × 1 × 4 convolution+ReLU max pooling fully connected+ReLU xeujos128 x 160 x 42 128 x 160 x 32 63 × 79 × 64 31x 39 x 128 7x9x256 5 x 19 x 256 1 × 1 × 1024 1 × 1 × 4 convolution+ReLU max pooling fully connected+ReLU softmax2 N Nin+NoutFigure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Validation accuracy obtained with different learning rates for the network trained on prostate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Validation accuracy obtained with different learning rates for the network trained on colorectal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Loss function evolution during training for the prostate data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' JMIR Bioinform Biotech 2022 | vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 3 | iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 1 | e27394 | p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 8 https://bioinform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='jmir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='org/2022/1/e27394 (page number not for citation purposes) Peyret et al JMIR BIOINFORMATICS AND BIOTECHNOLOGY XSL•FO RenderX 100 90 80 70 60 50 40 30 - TTTT TTT LL 10~7 10~6 10~s 10~4 10~3 TTTTTT 10~2 10~1 learning rate100 90 80 ccuracy 70 60 - 50 40 OE 10~7 TTTTT 10~6 TTTTTT 10~5 TTTT 10~4 TTTT 10~3 TTTTT 10~2 learning rateModel loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='75 train validation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='50 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='00 0 10 20 30 40 50 epochFigure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Loss function evolution during training for the colorectal data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Accuracy evolution during training for the prostate data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Accuracy evolution during training for the colorectal data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Transfer Learning Transfer learning consists of using a network previously trained on another data set to use the knowledge acquired during this learning task for the new task at hand [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' In most transfer learning for image classification tasks, the ImageNet data set [40], which contains 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='2 million images with 1000 categories, is used for pretraining the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' When only a small data set is available, this allows the CNN to be trained on a very large data set and therefore train a high-capacity network that captures details without overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Very deep networks also require a lot of time and very powerful machines equipped with multiple GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Using pretrained networks can be advantageous when appropriate resources are not provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Several transfer-learning scenarios are practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' In the first scenario, the pretrained CNN is used as a fixed feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The convolutional layers of the network are kept with the weights determined during training on the ImageNet data set, and the pretrained fully connected layers are replaced with fully connected layers initialized with random JMIR Bioinform Biotech 2022 | vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 3 | iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 1 | e27394 | p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 9 https://bioinform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='jmir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='org/2022/1/e27394 (page number not for citation purposes) Peyret et al JMIR BIOINFORMATICS AND BIOTECHNOLOGY XSL•FO RenderX Model loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='2 train 1.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='2 train 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='0 validation 0 10 20 30 40 50 epochModel accuracy 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5 train 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='4 validation 0 10 20 30 40 50 60 70 Epochweights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' During training, only the newly added fully connected layers were marked as trainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' They used the features extracted by pretrained convolutional layers as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' These features are usually referred to as CNN codes [26,39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Another strategy is to retrain the fully connected layers from scratch to fine-tune the weights of the pretrained convolutional layers by continuing back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Either all the convolutional layers can be retuned or only some of the higher-level layers to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This derives from the observation that the lower-level layers usually learn more generic features, such as edge detectors, that can be used for many different learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' In contrast, the high-level layers tend to learn features that are more specific to the characteristics of the classes of the original data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' In this study, only the first scenario was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The pretrained CNNs are very deep and require very high computational power to be retuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Using them as feature extractors is, in fact, equivalent to training only a relatively shallow MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The proposed architecture was compared with popular CNN architectures: VGG16 [37], InceptionV3 [21], and ResNet50 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' These networks were initialized with the weights obtained when pretraining them on the ImageNet data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, InceptionV3 and ResNet50 are very deep networks (48 and 152 layers, respectively), and a minimum input image size is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' InceptionV3 requires a minimum width and height of 139 pixels and ResNet50 of 197 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The images of the colorectal data set were smaller, and zero padding was added to reach the required dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Moreover, the ImageNet images are RGB images and therefore have a depth of 3 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' To meet the dimension requirements, a principal component analysis (PCA) was carried out to reduce the dimensionality of the multiscale images to 3 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Results and Discussion Principal Results and Findings To visualize the effect of the kernels on images through the network, Figures 11 and 12 present examples of outputs of the first convolutional layer of the networks trained with the prostate and colorectal data sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Figures 13 and 14 depict examples of outputs of the last convolutional layers of the same networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' It can be observed that after the first layer, the outputs are very similar to the input image, for instance, with transformations resembling edge detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Once the image has its own through the network, different regions or features of the input image are represented in the outputs of the last convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Thus, the different layers learn a succession of transformations, leading to an isolation of relevant regions or features of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The fully connected layers of the network are then able to classify these particular features into the 4 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Example of an output of the first convolutional layer for the network trained on the prostate data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' JMIR Bioinform Biotech 2022 | vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='25 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='25Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Example of an output of the last convolutional layer for the network trained on the colorectal data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Table 1 displays the validation and test accuracies obtained using the prostate and colorectal data sets for different CNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This shows that the validation and test accuracies are very close, proving a good generalization of the systems and that overfitting was avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The proposed CNN model achieved an average test accuracy of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='8% and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5% for the prostate and colorectal data sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Table 2 shows that the optimal CNN weights were obtained after 44 and 70 epochs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The VGG16 model initialized with Xavier weights trains very quickly for the prostate data set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' the optimal validation accuracy was obtained after 19 epochs, as illustrated in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, it is less efficient at learning for the colorectal data set and requires as many as 70 epochs to obtain the minimum validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The results also show slight overfitting for the colorectal data set, as the validation accuracy is lower than the training accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This is because of the high capacity of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' When using this network with pretrained weights from ImageNet, the training loss reaches a minimum after only a few epochs, but the validation loss shows that the network overfits marginally for both data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The test accuracy was also lower than that of the proposed CNN by 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5% and 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This is because the CNN codes learned with the ImageNet data set are not as adapted to the classification task at hand as those learned with the proposed CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The InceptionV3 model shows a higher overfitting and a lower generalization for both data sets with 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='0% and 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5% accuracy for the prostate and colorectal data sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This shows once again that the CNN codes learned on the ImageNet data set with this network are not adapted to the classification task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Finally, the pretrained ResNet50 achieved optimal accuracy with the lowest number of epochs: 5 and 22 for the prostate and colorectal data sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' It also achieves 100% average accuracy for the prostate data set, outperforming the proposed CNN, and 99% for the colorectal data set, which is slightly lower than the proposed data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This lower performance compared with the proposed CNN architecture for the colorectal data set might be owing to some loss of information when performing PCA on the 42 channels of the colorectal data set images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The prostate data set consisted of images with only 16 channels, and it is logical that the loss of information is not as important during this transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Therefore, the proposed CNN architecture is more adapted to the task at hand than the other methods it was compared with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, ResNet50 shows very good performance when used as a feature extractor and is trained with fewer epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' In every case, it was noted that the colorectal data set is more prone to overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This is probably owing to the size of the images, which are spatially smaller than those for the prostate data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Therefore, a model with the correct capacity for the prostate data set might be overestimated for the colorectal data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' JMIR Bioinform Biotech 2022 | vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 3 | iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 1 | e27394 | p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 12 https://bioinform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='jmir.' 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+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1000 100 0100010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='c 10 10 0 10 0 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Validation and test accuracy comparison of different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='a Colorectal data set (% accuracy), mean (SD) Prostate data set (% accuracy), mean (SD) Method Test Validation Test Validation 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1) 100 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1) 100 Proposed CNNb 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1) 100 VGG16c Xavier initial 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='2) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1) 100 VGG16 pretrained 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='3) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='3) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='2) InceptionV3 pretrain 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='2) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1) 100 100 ResNet50 pretrained aSD values have been provided wherever applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' bCNN: convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' cVGG16: Visual Geometry Group 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Number of epochs until early stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Colorectal data set Prostate data set Method 70 44 Proposed CNNa 70 19 VGG16b Xavier initialization 38 10 VGG16 pretrained 53 48 InceptionV3 pretrained 22 5 ResNet50 pretrained aCNN: convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' bVGG16: Visual Geometry Group 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Comparison Against Other Machine Learning Methods Table 3 shows the test accuracy of the best-performing CNN architectures compared with other methods from Tahir et al [15], Bouatemane et al [16], Haj-Hassan et al [25], and Peyret et al [17] stacked multispectral multiscale local binary pattern (MMLBP) + gray-level co-occurrence matrix (GLCM), and concatenated local binary pattern [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Regarding the prostate data set, 5 systems have an accuracy above 99%: Bouatemane et al [16], Stacked MMLBP+GLCM, the proposed CNN, Haj-Hassan et al [25], and ResNet50 with pretrained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The highest classification accuracy was achieved using ResNet50 with 100% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The proposed CNN and the study by Bouatemane et al [16] achieved 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='8% accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' however, the SD was not given for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Therefore, it is not possible to determine the precision of the accuracy estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The stacked MMLBP+GLCM system achieves 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5% (SD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='3 pp), which makes this performance similar to that of the proposed CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, a higher SD shows lower precision in the accuracy estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Therefore, the proposed CNN was preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The study by Haj-Hassan et al [25] achieved a 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='17% accuracy with segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Their system without this preprocessing phase achieved an accuracy of 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='23%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This can be explained by the lower capacity of their model compared with ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This has the advantage of requiring less processing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, this is counterbalanced by the fact that their system requires a preprocessing phase with the intervention of a pathologist, which dramatically increases the processing time of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Furthermore, they state that their CNN model requires 500 epochs to be trained, which is much higher than that of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' With respect to the colorectal data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Peyret et al [17] stacked MMLBP+GLCM system and the proposed CNN both provided the same accuracy and SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' They outperform ResNet50 with pretrained weights by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Finally, when considering the results obtained with both data sets, the stacked MMLBP+GLCM system and the proposed CNN appear to provide the most stable results as well as the highest accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, on average, the SD of the accuracy achieved by the proposed CNN is lower than that obtained with the stacked MMLBP+GLCM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The performance of the ResNet50 network seems to be more dependent on the data set used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Moreover, it would be interesting to compare the system proposed by Bouatemane et al [16] using the colorectal data set to verify whether it performs as well on different data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Considering the current information available on the system performance and with the data sets available, the proposed CNN is selected as the best-performing system in terms of accuracy for the classification task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' JMIR Bioinform Biotech 2022 | vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 3 | iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 1 | e27394 | p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 13 https://bioinform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='jmir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='org/2022/1/e27394 (page number not for citation purposes) Peyret et al JMIR BIOINFORMATICS AND BIOTECHNOLOGY XSL•FO RenderX Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Accuracy comparison against other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='a Colorectal data set (% accuracy), mean (SD) Prostate data set (% accuracy), mean (SD) Method Tahir et al [15] N/Ab 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='9 Bouatemane et al [16] N/A 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='83 Concatenated LBPc [18] 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='4) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='3) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1) Stacked MMLBPd + GLCMe [41] 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='3) Haj-Hassan et al [25] N/A 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='17 Proposed CNN 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1) ResNet50 pretrained 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='2) 100 aSD values have been provided wherever applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' bN/A: not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' cLBP: local binary pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' dMMLBP: multispectral multiscale local binary pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' eGLCM: gray-level co-occurrence matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Computational Complexity Analysis In computer-aided diagnosis systems (CADSs), an unlabeled image is fed to a previously trained system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Consequently, the time used to process this image is decisive, as it is crucial that the CADS works on the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, the forward pass of an image through the CNN architectures studied in this study is computationally nonexpensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Table 4 displays the classification times per image for all CNN architectures tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This demonstrates that only a few milliseconds are required to classify one image once the CNN has been trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, it must be noted that the proposed CNN architecture is much quicker at classifying images than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This is because, for the architectures described in the literature and the pretrained networks, a PCA must be carried out to reduce to 3 the number of channels of the image to be classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' This preprocessing stage lengthens the total classification time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' As mentioned, training is performed only once when a CADS is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Consequently, training time is not a critical measure of the problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, the computational complexity of deep learning systems can rapidly increase significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Such architectures require high-performing hardware, including GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Some extremely deep architectures can also entail several weeks of training time [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Such long training times considerably slowed down the CADS development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' To verify that the proposed system can be trained within a reasonable duration, a comparison of the training times for each architecture was carried out (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The computational times depending on the hardware and software used, it is not possible to compare the CNN architectures with other classification systems proposed in other published works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, this is one of the first attempts to use deep learning for this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Therefore, this section aims to establish the ability of deep learning systems to be trained in a short period using the data sets used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Unsurprisingly, Table 5 demonstrates that pretrained networks have a much shorter training time per epoch owing to the reduced number of layers to be trained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' ResNet50 and InceptionV3 can be trained in a few minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' When considering this measure of performance, the best architecture was ResNet50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' However, the total training time for every CNN model is <2 hours, making it a reasonable time for developing a CADS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Average convolutional neural network (CNN) classification computation times for 1 image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Colorectal data set (ms) Prostate data set (ms) Method 7 14 Proposed CNN 42 75 VGG16a Xavier initial 42 75 VGG16 pretrained 42 63 InceptionV3 pretrained 47 65 ResNet50 pretrained aVGG16: Visual Geometry Group 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' JMIR Bioinform Biotech 2022 | vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 3 | iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 1 | e27394 | p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' 14 https://bioinform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='jmir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='org/2022/1/e27394 (page number not for citation purposes) Peyret et al JMIR BIOINFORMATICS AND BIOTECHNOLOGY XSL•FO RenderX Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Average convolutional neural network (CNN) training computation times for the complete data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Colorectal data set (seconds) Prostate data set (seconds) Method Total training Time per epoch Total training Time per epoch 2925 45 3780 90 Proposed CNN 6790 97 4655 245 VGG16a Xavier initial 1400 35 3154 83 VGG16 pretrained 705 15 1755 39 InceptionV3 pretrained 704 32 205 41 ResNet50 pretrained aVGG16: Visual Geometry Group 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Conclusions In this paper, the proposed CNN architecture was detailed and compared with previously trained network models used as feature extractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' These CNNs were also compared with other classification methods from other published studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The proposed CNN demonstrated excellent performance compared with pretrained CNNs and other classification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The computational complexity of the CNNs was also analyzed, and it was demonstrated that the proposed CNN is faster at classifying images than pretrained networks because it avoids a preprocessing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' The conclusion of this overall analysis is that the proposed CNN architecture was globally the best-performing system for classifying colorectal and prostate tumor images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Acknowledgments This research project was supported by a grant from the Research Supporting Program (Project Number: RSP2022R281), King Saud University, Riyadh, Saudi Arabia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Conflicts of Interest None declared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' ImageNet large scale visual recognition challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Int J Comput Vis 2015 Apr 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='115(3):211-252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' [doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1007/s11263-015-0816-y] 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Peyret R, Bouridane A, Khelifi F, Tahir M, Al-Maadeed S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Automatic classification of colorectal and prostatic histologic tumor images using multiscale multispectral local binary pattern texture features and stacked generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' Neurocomputing 2018 Jan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='275(C):83-93 [FREE Full text] [doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='neucom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='010] Abbreviations CADS: computer-aided diagnosis system CNN: convolutional neural network GLCM: gray-level co-occurrence matrix GPU: graphic processing unit MLP: multilayer perceptron MMLBP: multispectral multiscale local binary pattern PCA: principal component analysis ReLU: rectified linear unit RGB: red, green, blue VGG16: Visual Geometry Group 16 Edited by A Mavragani;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' submitted 23.' metadata={'source': 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version received 08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' accepted 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content='21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} +page_content=' published 09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFPT4oBgHgl3EQftDXh/content/2301.13151v1.pdf'} 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Cristea a, b, Derrick G. Watson c +a Department of Computer Science, Durham University, Durham, DH1 3LE, United Kingdom +b Department of Computer Science, University of Warwick, Coventry, CV4 7AL, United Kingdom +c Department of Psychology, University of Warwick, Coventry, CV4 7AL, United Kingdom + +Abstract: The effect of emotions and personalisation on continuance use intentions in online health +services is underexplored. Accordingly, we propose a research model for examining the impact of +emotion- and personalisation-based factors on cancer website reuse intentions. We conducted a +study using a real-world NGO cancer-support website, which was evaluated by 98 participants via +an online questionnaire. Model relations were estimated using the PLS-SEM method. Our findings +indicated that pre-use emotions did not significantly influence perceived personalisation. However, +satisfaction with personalisation, and perceived usefulness mediated by satisfaction, increased +reuse intentions. In addition, post-use positive emotions potentially influenced reuse intentions. Our +paper, therefore, illustrates the applicability of theory regarding continuance use intentions to +cancer-support websites and highlights the importance of personalisation for these purposes. +Keywords: cancer website; continuance intention; emotions; perceived usefulness; +satisfaction with personalization + +1. Introduction +The demand for digital health has been rising 1, 2. However, challenges and barriers to its +usage have also become more evident. A number of prior studies, thus, advocated for +personalisation 3, i.e., tailoring content to user needs, to improve engagement with digital +health systems 2, and increase user satisfaction and usage intentions 4. +Online health service personalisation has been explored from the perspective of: its +association with self-disclosure 5, systematic literature reviews which have highlighted its +benefits 6, personalised educational health content for the elderly 7 and adolescents 8, +distress-based personalised therapy recommendations 9, and recommending health and +fitness content for runners 10. Nevertheless, research exploring the different factors that +might influence intentions to (re)use personalised online health services is lacking. Our +study addresses this gap, specifically for cancer-support websites. +Moreover, much has been discovered about how information technology (IT) use intentions + +2 + +are affected by cognitive factors, as are perceived usefulness and satisfaction 11-15. +However, there is insufficient understanding of how a user’s affective states might +influence their use of online personalisation 16, 17, although it is known that emotions impact +human behaviour and perception 18, and are induced by different interactions 19. Therefore, +there is good reason to explore if emotions can lead to and/or be influenced by the use of +personalised online health services. +Indeed, persistent sadness or anxiety were shown to increase the likelihood of +online health information seeking 20. Some studies examined the effects of website design +21, interface aesthetics 22 and website features 23 on affective responses (e.g., arousal and +irritation). Furthermore, the possible links between emotions and online personalisation +have been considered in the domains of: e-commerce 4, group decision support systems 24, +e-learning 17 and emotion-aware recommender systems 9, 25, 26. However, to the best of our +knowledge, our study is the first to propose a research model that explores emotions and +perception of personalisation as factors influencing continuance use intentions in the +context of cancer-support websites. +2. Theoretical background and research model + +2.1. Underlying theories +The underlying theories for our conceptual framework were: i) the two-stage model of +cognition change toward IT usage 27 (Figure 1, part 1), ii) affect theory (Figure 1, part 2) , +which defines emotions as drivers of human behaviour 18, iii) and appraisal theories +(Figure 1, part 3), which describe emotions as reactions 19 to the current context’s +assessment 28, 29. The two-stage model has previously been applied to, e.g., digital learning +technologies 15, however, not to personalised cancer-support websites. The model measures +changes in beliefs and attitude from pre- to post-usage stages, and satisfaction as a post- +usage affective state. +Our conceptual framework integrates the belief- and affect-based constructs at different IT +use stages, however, we only examine changes in affective state. Furthermore, we measure +belief only at the post-use stage, specifically as the perceived usefulness of personalisation. +In the two-stage model, the satisfaction construct captures the extent of user satisfaction, +pleasantness, content, and delight toward IT. We, on the other hand, use this construct to +represent post-use cognitive-based appraisal of different aspects of personalisation (Section +2.2.4 and Table A1). + +3 + + +Figure 1. Conceptual framework - expanded two-stage model of IT usage intentions with emotions +Importantly, the two-stage model provides only a limited understanding of the relationship +between emotions and perceptions or behavioural intentions, i.e., via satisfaction. Pre- and +post-usage attitudes measure user perception (i.e., cognitive appraisal) of whether system +use was good, wise, positive and effective. In contrast, our pre- and post-use emotions’ +factors explicitly capture user emotions, which are a very specific affect type (Section +2.2.2). Hence, our framework extends the two-stage model. It incorporates emotion-specific +constructs, which were derived from affect theories 18, 19 and measured by user self- +assessment of basic emotions’ intensities experienced at a particular moment of system use. +2.2. Research model and hypotheses +Building upon the conceptual framework from Section 2.1, Figure 2 illustrates our research +model. This section defines its constituting factors and hypotheses. + + +Figure 2. Research model + + +Pre-use stage +During-/post-use stage +Affective state +Cognitive appraisal +Affective appraisal +Positive +Positive +Perceived +Reuse +emotions +emotions +usefulness +intentions +Negative +Satisfaction +Negative +emotions +emotions +Two-stage model of IT usage intentions +1 +2 +Affect theory +Affect theory +3 +Appraisal theoryPre-use stage +During-/ post-use stage +Satisfaction +with +Pre-use +personalisation +positive emotions +Usefulness +Reuse +of +intentions +Pre-use ++ +Post-use +personalisation +negative emotions +positive emotions +Post-use +negative emotions4 + +2.2.1. Reuse intentions +Behavioural intention is an intent to perform certain behaviour 1. Theories of reasoned +action and planned behaviour introduced this concept, followed by its adoption in IT use 30 +and continuance intentions 27 frameworks, and applications in, e.g., online personalisation 4, +31 and online health services 1, 32, 33. Our study measures users’ intentions to revisit and use +a personalised cancer-support website. +2.2.2. Emotions (pre- and post-use) +Emotions are high intensity, brief affective states 24 that begin quickly 34. We measured 12 +emotions, selected based on the results from two prior studies. The first study 35 explored +the likelihood that interest, fear, sadness, surprise, awe, anger, embarrassment, guilt, +enjoyment, shame, happiness, contempt, or disgust stimulate online cancer information +searching. The second study 36 examined the association between online personalisation +needs, usage intentions, and 13 basic and possible basic emotions, as classified by 34, 37. +These two studies showed that only certain basic emotions play a significant role in online +health information use or perceived personalisation needs. These were fear, sadness, awe, +excitement, interest and surprise, and were hence all accounted for in our study. We also +added six other emotions. Anxiety, boredom and calmness (or neutral state) were included +due to their frequent use in the related human-computer interaction (HCI) research 16, 38-41. +Embarrassment, guilt and happiness were basic emotions that were re-evaluated in this +study to balance the number of positive and negative emotions measured. Happiness (or +alternatively joy, enjoyment, pleasantness) is also one of the essential positive valence +emotions often studied in HCI 42. Based on vulnerability research 43, embarrassment and +guilt were included to represent the cancer-affected people’s potential negative perception +of their own self, state or actions. +Given that positive and negative emotions influence behaviour differently 18, we used +previous research 29, 34, 44-47 to classify the 12 emotions into positive (denoted above in +italics) and negative valence categories. Furthermore, we measured emotions at two stages +(Section 2.1): pre-use emotions at the beginning of website use, and post-use emotions after +website use. +Emotions as stimuli of perception or action +The effect of emotions on IT perception has been addressed in a few studies. For example, +emotional attachment influenced the perceived usefulness and attitude towards Facebook 48, +and affective quality of smart watches was associated with their perceived usefulness 13. +However, such research in the online health domain is very limited, e.g., indicating that + +5 + +interest and excitement increased the perceived needs for personalisation 36. Given the +under-researched association between emotions and perceived personalisation, we +hypothesised that: +H1.1: Pre-use positive emotions increase the perceived usefulness of personalisation. +H1.2: Pre-use negative emotions decrease the perceived usefulness of personalisation. +Interestingly, there is more research on the association between emotions and behavioural +intentions towards IT. Enjoyment was found to influence web use 41, anxiety influenced +continuance use intention for mobile-health among elderly 49, emotions stimulated online +search for cancer information 35, and interest increased cancer patients’ use of +electronic health records 50. Strong positive emotions, or absence of negative emotions, +mediated the effect of personalisation on online purchase intentions 51. However, we found +only one prior study 36 that used linear regression and showed a positive influence of +interest on reuse intentions of partially personalised online cancer services. +Due to the lack of research on cause-effect relations (or structural equation modelling) +between emotions and user intentions on personalised cancer-support websites we +hypothesised that: +H2.1: Post-use positive emotions increase reuse intentions. +H2.2: Post-use negative emotions decrease reuse intentions. +Emotions as reactions or affective appraisal +We argue that users’ appraisal of cancer-support website personalisation can evoke post- +use positive and negative emotions. This is based on research showing that certain website +features (e.g., colour, images, shapes) induced emotions 52, perceived usefulness of +educational blogs increased liking and pleasantness 11, pedagogical agent’s adaptation +intensified enjoyment and decreased boredom 42, personalisation predicted post-use positive +emotions in online shopping 51, and online health information overload (i.e., lacking +personalisation) influenced negative emotions +53. However, there is insufficient +understanding about the impact of personalised online health services 52 on affective states. +We therefore postulated the following hypotheses: +H3.1: Perceived usefulness of personalisation positively influences post-use positive +emotions. +H3.2: Perceived usefulness of personalisation negatively influences post-use negative +emotions. +Moreover, consistent with research that showed that satisfaction positively influenced +attitude after using IT 27, 54, we hypothesised that: + +6 + +H4.1: Satisfaction with personalisation positively influences post-use positive emotions. +H4.2: Satisfaction with personalisation negative influences post-use negative emotions. +2.2.3. Usefulness of personalisation +Perceived usefulness relates to expectations about performance improvements as a result of +using a service or product 54. Our usefulness of personalisation factor adapted Davis’s 55 +‘perceived usefulness’ to evaluate the individual personalisation features implemented on +the studied cancer-support website. This approach, has also been applied to e-commerce 44 +and personalised e-learning 56. +Previous work has shown that perceived usefulness had a significant positive impact on +satisfaction in the use of, e.g., online- and m-banking 57, 58, e-government 54, and digital +textbooks 15. Moreover, personalisation applied to online news 59, e-commerce 60 and +online-banking 61 increased user satisfaction. Therefore, for cancer-support websites, we +hypothesised that: +H5: Perceived usefulness of personalisation increases satisfaction with personalisation. +Furthermore, previous research has argued that perceived usefulness 55 is an essential +criterion for system reuse 62. It increased usage intentions for digital textbooks 15, health- +information portals 32 and mobile health applications 63. Hence, we hypothesised that: +H6: Perceived usefulness of personalisation increases reuse intentions. +2.2.4. Satisfaction with personalisation +In a seminal work on customer expectations, Oliver 64 defines satisfaction as the “summary +psychological state resulting when the emotion surrounding disconfirmed expectations is +coupled with the consumer’s prior feelings about the consumption experience”. Abundant +research shows that satisfaction positively affects online repurchase intentions 65 and +(continuance) usage intentions for online-banking 57, e-government 54, educational blogs 11, +m-health 33, and personalised information reuse 60. Accordingly, with respect to +personalised cancer-support websites, we hypothesised that: +H7: Satisfaction with personalisation increases reuse intentions. + + + + +7 + +3. +Methodology + +3.1. Study design and data collection +We sampled people directly and indirectly affected by cancer, i.e.: former/current patients; +caregivers – family and friends; and those preventatively seeking cancer information. This +was achieved by using: + purposive sampling for cancer association members; + and convenience sampling for university students (as primary users of online health +services 66), social networks’ users, and crowdsourced participants (via Amazon +Mechanical Turk1). +This study was reviewed and approved (REGO-2015-1421) by the Biomedical and +Scientific Research Ethics Committee at the University of Warwick. It used an online +survey methodology. The survey first explained the study’s objective, the right to withdraw +without consequences, and informed participants they were consenting to take part in this +research and the collection of their anonymised responses. The following were then +presented in a consecutive order: +(1) Questionnaire: Reporting pre-use emotions (≈ 5 minutes). +(2) Experiment: User-website interaction (≈ 25 minutes). +(3) Questionnaire: Post-use emotions and website evaluation (≈ 30 minutes). + +Figure 3. PORT cancer website + +1 https://www.mturk.com/ + +PORT +WELCOME,SUNGM +MYPORT +NGO helping cancer patients in Bosnia and +Herzegovina +ABOUTUS +ARTICLES +KNOWLEDGEBASE +VIRTUALCOMMUNITY +Emotiontool +MYPROFILE +BLOG +READLIST +USERSFOLLOWED +ACTIVITIES +emotions,andtowhichextent? +★WHATDIDYOU THINK ABOUTTHEFOLLOWING CONTENT? +Fear +Immunotherapy-'Cureforterminal cancer found in game- +Interest +changing drugs +Sadness +Surprise +Name:sungm +Avwe +RECOMMENDEDCONTENT +EDIT +Happiness +Embarrassment +OCHATROOM +Excitement +3/5 +KNOWLEDGEBASE +Guit +21/03/1614:37 +Cancerassociationsin Bosniaand Herzegovina +Anxiety +Is this a use +Boredom +Calmness +KNOWLEDGE BA SE +04/03/1622:05 +Why does cancer reoccur?8 + +During the experiment (step 2), participants interacted with PORT 67, a personalised cancer- +support website (Figure 3) for patients and caregivers. PORT’s cancer-related content +included cancer patients’ blogs, and articles adopted from respectable online sources about +different cancer types, treatments and therapies. Participants completed the following2: +user-profile creation; privacy policy customisation; user-profile editing; interface +adaptation +(e.g., +adjusting +font, +colour, +etc.); +rating content +and +reviewing +recommendations. Since we were interested in the effect of interaction with a personalised +cancer-support website, these tasks were essential for a user to explore the website, receive +and perceive the personalisation, which on PORT comprised cancer information +recommendations and user interface adaptation. The questionnaires (steps 1 and 3; see +Appendix A) collected data on pre-use emotions, user demographics, post-use emotions, +perceived usefulness of and satisfaction with personalisation, and website reuse intentions. +The scale for measuring emotion intensity was adopted from a game experience +questionnaire 68, applied to online systems 44, 69. Items from validated instruments were +used for satisfaction with personalisation 70, 71 and reuse intentions 51, 72. The perceived +usefulness instrument +55 was modified to measure the usefulness of individual +personalisation features implemented on the PORT website (see Appendix A). +The online survey started in May 2015, and ran for 1.5 months. We received 122 responses; +98 were valid and used in data analysis. We removed the data from respondents who were +not affected at all or not interested in cancer. +3.2. Data analysis and instrument validation +Data pre-processing, exploratory factor analysis (EFA) and descriptive analyses were +conducted using IBM SPSS® Statistics3. SmartPLS 34 was used for confirmatory factor +analysis (CFA) and structural equation modelling (SEM) with partial least squares (PLS) +method. +EFA was only applied to the 24 items for usefulness of personalisation (Appendix A), as +we modified the original instrument. We used principal axis factoring 73, direct Oblimin +rotation 74, with Kaiser normalisation and a fixed number of factors based on our previous +studies, which were confirmed with Eigenvalues>1.0 and a scree test 73. A two-factor +solution was selected: 55.89% variance explained; Eigenvalues>1.43; KMO = 0.76; χ2(55) += 440.03, p < 0.001. 7 items reflected the factor usefulness of content-related +personalisation (UsfCP), and 4 items represented the factor usefulness of explicit UI- and + +2 See Appendix A for the complete list of website features participants were exposed to, asked to interact with and +evaluate on perceived usefulness. +3 https://www.ibm.com/uk-en/products/spss-statistics +4 https://www.smartpls.com/ + +9 + +content-adaptation (UsfADP). Namely, UsfCP covers the automatically generated +recommendations of different content (e.g., articles, blog posts) and the content rating +functionality required for these purposes. UsfADP, on the other hand, refers to website +features requiring more explicit user involvement for content customisation (e.g., +notifications and privacy policy length) and text appearance adaptation. +We next ran a CFA in SmartPLS on the refined model with eight factors: pre-use positive +(PREPE) and negative emotions (PRENE), post-use positive (POPE) and negative +emotions (PONE), usefulness of content-related personalisation (UsfCP), usefulness of UI- +/content-adaptation (UsfADP), satisfaction (SAT) and reuse intentions (RI). Cronbach’s α +and composite reliability ≥0.7 75 and AVE - average variance extracted >0.5 76 were +achieved by iteratively removing items with low outer loadings - starting with <0.5, up to +0.7 73, 77. Table 1 presents reliability and validity results, and Table A1 (Appendix A) factor +loadings. The Fornell-Larcker criterion for discriminant validity was satisfactory 78. +Heterotrait-monotrait ratio 79 was <0.85 for all factors; apart from pre-use to post-use +negative emotions (HTMT = 0.907), likely due to the same constituting emotions: fear, +sadness and guilt/embarrassment). However, the latter was acceptable at the HTMTinference +criterion 79. +Table 1. Construct reliability and validity +Factor +Num. of +items +Mean (SD); N +Cronbach's α +Composite +Reliability +AVE +PREPE +2 +1.6 (0.8); 98 +0.731 +0.734 +0.581 +PRENE +3 +1.6 (0.8); 98 +0.771 +0.771 +0.529 +UsfCP +4 +3.9 (0.7); 97 +0.826 +0.828 +0.546 +UsfADP +3 +3.8 (0.8); 97 +0.757 +0.754 +0.507 +SAT +3 +3.9 (0.7); 96 +0.797 +0.796 +0.566 +POPE +2 +1.8 (0.9); 98 +0.734 +0.734 +0.580 +PONE +3 +1.7 (0.8); 98 +0.761 +0.763 +0.520 +RI +3 +3.7 (0.8); 98 +0.820 +0.820 +0.604 +4. Results + +4.1. Participant demographics +The respondents' age ranged from 18 to 57 years (Mean=27, SD=8.9). The majority were +from B&H (51%) and USA (33.7%), and 61.2% were female. They were mainly caregivers +to a family member who suffered from cancer (54.1%), preventatively sought cancer +information (30.6%), had a friend suffering from cancer (14.3%), or were a cancer patient +(1%). + +10 + +4.2. PLS-SEM results +Model fit was tested with a consistent PLS algorithm - all LVs connected for initial +calculation, 300 iterations, path weighting scheme, missing values replaced with a mean. +SRMR (0.069 *saturated, 0.181 **estimated model) met the recommended value of <0.08 +80, while NFI (0.717*, 0.587**) was slightly below the recommended 0.9-1 81. Figure 4 +shows the path coefficients (ß) and coefficients of determination (R2) after applying +complete bootstrapping with 2000 subsamples. + +Figure 4. Estimated model - path coefficients and significance levels +The findings showed that four path coefficients were significant at p<0.05, supporting +H4.1, H5, H7. At the pre-use stage, negative emotions (specifically fear, guilt and sadness +categories) decreased the usefulness of adaptation-related personalisation, however at p<0.1 +(H1.2: ß = -.19, t = 1.71, p = .088). During website use, perceived usefulness of content +personalisation (H5 - UsfCP: ß = .32, t = 2.93, p = .003) and adaptation (H5 - UsfADP: ß = +.49, t = 4.77, p = .000) significantly increased satisfaction. However, without a direct effect +on post-use emotions or reuse intentions (H3.1, H3.2, H6 were not supported). +At the during- and post-use stage, satisfaction with personalisation intensified positive +emotions (H4.1: ß = .44, t = 3.2, p = .001). Satisfaction (H7: ß = .45, t = 3.6, p = .000), and +potentially post-use positive emotions (H2.1: ß = .17, t = 1.7, p = .090), increased reuse +intentions. Interestingly, post-use negative emotions did not influence and were not +influenced by the factors in our model (H2.2 and H4.2 not supported). +We also tested mediating effects (Table 2), based on Zhao’s method 82. Post-use emotions +were not significant mediators. Nevertheless, satisfaction fully mediated the effect of +usefulness of content- and adaptation-related personalisation (UsfCP and UsfADP, +respectively) on post-use positive emotions and reuse intentions. + + +Pre-use stage +I During-/ post-use stage +Satisfaction +with +personalisation +H7 +H5 +(R2=.486) +0.45** +0.32** +H5 +H4.1 +H1.1 +Pre-use +UsfCP +H6 +Reuse +-9.21 (n.s.) +0.49** +0.44** +positive emotions +Usefulness of +-9.001 (n.s.). +intentions +H1:1 +personalisation +(R2=.426) +-0.08 (n.s +H3.1 +H6 +H2.1 +(R2=.036) +-0.05 (n.s) +0.13 (n.s.) +0.17# +H1.2, +0.11 (n.s.) +Post-use +Pre-use +UsfADP - +H3.1 + positive emotions +Usefulness of +negative emotions +-0.23 (n.s.) +H1.2 +(R2=.110) +personalisation +H4.2 +/H2.2 +-0.19# +H3.2 +I 0.14 (n.s.) +(R2=.054) +0.21 (n.s.) +0.12 (n.s.) +Dashed line / n.s.: not significant +H3.2 +#p<.1 +-0.23 (n.s.) +* p<.05 +Post-use +** p<.01 +negative emotions +(R2=.061)11 + +Table 2. Mediating effects +IV +M +DV +P1: +IV->M +P2: +M->DV +P3: +IV->DV +P1⋅P2 +Result +UsfCP +SAT +POPE +0.32** +0.44** +n.s. +0.14# +Full mediation +UsfCP +SAT +RI +0.32** +0.45** +n.s. +0.14* +Full mediation +UsfADP SAT +POPE +0.49** +0.44** +n.s. +0.22** +Full mediation +UsfADP SAT +RI +0.49** +0.45** +n.s. +0.22** +Full mediation +IV: +independent +variable, +M: +mediator, +DV: +dependent +variable. +# p < 0.1; *p < 0.05; **p < 0.01 +5. +Discussion +Our findings imply that the essential factor explaining reuse intentions for cancer-support +websites is satisfaction with personalisation. It mediates the effect of usefulness of +personalisation, and directly increases reuse intentions, as seen in numerous studies on +continuance use intentions in other domains 11, 15, 57. We next discuss and generalise the key +results. +First, pre-use emotions do not significantly affect perceived personalisation. Although prior +online-health research indicated a possible effect of positive emotions on personalisation +needs 36, our study showed that surprise and awe (i.e., positive-valence, high-arousal +emotions) do not influence usefulness of personalisation. Furthermore, we found a +marginally significant effect of negative emotions, such that fear, guilt and sadness jointly +decrease the usefulness of explicit UI- and content-adaptation (UsfADP), i.e., the type of +personalisation which requires explicit user involvement. This likely occurs because people +in negative affective states are biased towards negative events/occurrences 83, hence might +not perceive the benefits of personalisation. Overall, these are valuable findings, providing +an insight into the online cancer-support context, and inviting exploration of alternative +emotion taxonomies and their association with perceived personalisation. +Second, contrary to our prediction, usefulness of personalisation does not directly impact +post-use emotions. However, usefulness of personalisation (i.e., both content-related +personalisation and explicit UI- and content-adaptation) intensifies post-use positive +emotions when mediated by satisfaction. Our results are consistent with e-commerce +research regarding negative emotions 51, however, there, personalisation increased positive +emotions 51. This difference could stem from the different measurement methods: we +observed discrete emotions, while Pappas et al. 51 measured positive or negative mood; we +evaluated the perceived usefulness of individual personalisation features, and they +examined users’ willingness to be provided with personalisation. + +12 + +Third, cognitive perception of personalisation is more important than its affective appraisal +31. Almost 50% of variation in satisfaction with personalisation is explained by the +personalisation’s perceived usefulness. Thus, our findings align with the positive effect +found in online banking 57, 58, e-government 54 and digital textbooks 15. While research has +addressed the effect of satisfaction on attitude 27, 54, our study was the first to explore its +influence on post-use emotions. Specifically, we found that satisfaction with +personalisation intensifies post-use positive emotions, indicating a pleasant surprise after +confirming positive or disconfirming negative expectations. +Finally, contrary to the findings of prior research in other domains, reuse intentions for +personalised cancer-support websites are not significantly explained by post-use negative +emotions or perceived usefulness. Post-use negative emotions and benefits of +personalisation affected online purchase intentions in 51, and negative affects, depressive +symptoms and trait anger reduced online health information search intentions in 53. Thus, +behavioural intentions are possibly context- or task-dependent, or influenced differently by +various affective states. In fact, our findings suggest that post-use surprise and awe could +increase cancer website reuse intentions, which aligns with the findings for positive +emotions in, e.g., online purchasing 31, 51. +6. +Conclusion +From a theoretical perspective, our research implies that the two-stage model’s constructs - +usefulness and satisfaction - were applicable to understanding continuance use intentions +for personalised cancer-support websites. However, alternative theories, e.g., Theory of +Constructed Emotion 84, should be used for investigating the cause-effect between emotions +and personalisation. +Unlike the theory-proposed effect 18, emotions in the cancer-support website context were +not a significant predictor of perceived personalisation or behavioural intentions. +Nevertheless, we confirmed that context appraisal 28, 29, i.e., perceived personalisation did +evoke emotions. Furthermore, the frequently reported: i) effect of perceived usefulness on +satisfaction with IT, and ii) the influence of satisfaction with IT on its reuse intentions, also +prevail in cancer-support websites. Our study’s important contribution was measuring +perceived usefulness and satisfaction in relation to personalisation. +Furthermore, this paper offers practical implications. Cancer-support website providers +should implement personalisation, particularly content recommendations and interface +adaptation. These features increase satisfaction and positive emotions, hence stimulate +website reuse. + +13 + +Our research, however, has limitations. Although comparable to computer-use studies 17, 85, +86, our sample size was relatively small. The sampled participants here were mainly people +indirectly affected by cancer; future research should focus on cancer patients. Our findings’ +generalisability is currently limited to cancer-support websites. Moreover, alternative +emotion taxonomies could be examined and longitudinal studies for a deeper insight into +perceptions of personalisation. +In conclusion, this research uniquely applied affect and IT usage theories. Finally, its main +contribution is highlighting the importance of the understudied factors – emotions and +personalisation - in forming user intentions toward online cancer-related services. +Appendix A +Table A1. Overview of questionnaire items, measurement scales, factors and factor loadings +Factor +Questionnaire items +Outer +loadings +5-point scale: 1: Not experiencing this emotion at all, 2: Mildly, 3: Moderately, 4: Very, 5: +Experiencing this emotion extremely +Pre-use positive emotions +Awe +0.807 +Surprise +0.715 +Calmness, Excitement, Happiness, Interest (removed) +<0.7 +Pre-use negative emotions +Guilt +0.738 +Fear +0.734 +Sadness +0.709 +Anxiety, Boredom, Embarrassment (removed) +<0.7 +Post-use positive emotions +Surprise +0.762 +Awe +0.761 +Calmness, Excitement, Happiness, Interest (removed) +<0.7 +Post-use negative +emotions +Embarrassment +0.814 +Sadness +0.692 +Fear +0.647 +Anxiety, Boredom, Guilt (removed) +<0.64 +5-point scale: 1: strongly disagree to 5: strongly agree +Usefulness of … (I +perceive as useful the +personalisation feature…) + +…content-related +personalisation (UsfCP) + +UsfCP1. Recommendations for forum discussions +0.777 +UsfCP2. Recommendations for blogs +0.750 +UsfCP3. Recommendations for articles/news +0.723 +UsfCP4. Content rating +0.704 +UsfCP5. Recommendations for knowledge-base content +(removed) +<0.7 +UsfCP6. Personal readlist (removed) +<0.7 + +14 + + +UsfCP7. Categorising content (popularity, recency, etc.) +(removed) +<0.7 +…explicit UI- and content- +adaptation (UsfADP) +UsfADP1. Privacy policy customisation (long/concise) +0.749 +UsfADP2. Notifications/reminders +0.727 +UsfADP3. Text size adaptation +0.657 +UsfADP4. Text colour adaptation (removed) + <0.65 +…other evaluated +personalisation features +(removed after EFA) +Tailoring +background +colour; +User-profile +customisation; Defining interests; Feedback about +personalisation usefulness; “What did you think about +this +content?”; +Emotion +tool; +Filtering +search; +Recommendations +matching +user’s +interests; +Recommendations based on ratings; Recommendations +based on user similarity; Filtering recommendations; +Customising language; Greetings with username + +Satisfaction with +personalisation +(I am satisfied with how +PORT’s website was +personalised to my needs +because it…) +SAT1. … provided content at the right level of detail +0.793 +SAT2. … provided valuable content to me +0.751 +SAT3. … could save me time +0.710 +SAT4. … knew what I wanted (removed) +<0.7 +SAT5. … took into consideration my interests and +preferences to make recommendations to me (removed) +<0.7 +SAT6. +… +improved +my +search +performance +(removed) +<0.7 +SAT7. … provided relevant content to me (removed) +<0.7 +SAT8. … provided up-to-date content to me +(removed) +<0.7 +Reuse intentions +RI1. Overall, I have a positive attitude toward using the +website. +0.839 +RI2. Given the chance, I intend to use the website again +0.744 +RI3. I would recommend the website to my friends. +0.744 +RI4. I intend to use the website frequently. (removed) +<0.7 +Funding +This research received no specific grant from any funding agency in the public, +commercial, or not-for-profit sectors. +Declaration of conflicting interests +None to declare. + +15 + +References +1. +Hong Z, Deng Z and Zhang W. Examining factors affecting patients trust in online healthcare +services in China: The moderating role of the purpose of use. Health informatics journal 2018: +1460458218796660. +2. +Kim M YJ, Ahn WY, Choi HJ. Machine Learning Analysis to Identify Digital Behavioral +Phenotypes for Engagement and Health Outcome Efficacy of an mHealth Intervention for +Obesity: Randomized Controlled Trial. Journal of medical Internet research 2021; 23(6): +e27218. +3. +Sillence E, Little L and Briggs P. E-health. 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Journal of Computers 2013; 8. + + diff --git a/bNAyT4oBgHgl3EQf9_rY/content/tmp_files/load_file.txt b/bNAyT4oBgHgl3EQf9_rY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..545ff4dbba572c0c87d003fd02f3d9cfcf5f6c2b --- /dev/null +++ b/bNAyT4oBgHgl3EQf9_rY/content/tmp_files/load_file.txt @@ -0,0 +1,1033 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf,len=1032 +page_content='1 Effect of emotions and personalisation on cancer website reuse intentions Sunčica Hadžidedić a, b, Alexandra I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Cristea a, b, Derrick G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Watson c a Department of Computer Science, Durham University, Durham, DH1 3LE, United Kingdom b Department of Computer Science, University of Warwick, Coventry, CV4 7AL, United Kingdom c Department of Psychology, University of Warwick, Coventry, CV4 7AL, United Kingdom Abstract: The effect of emotions and personalisation on continuance use intentions in online health services is underexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Accordingly, we propose a research model for examining the impact of emotion- and personalisation-based factors on cancer website reuse intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' We conducted a study using a real-world NGO cancer-support website, which was evaluated by 98 participants via an online questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Model relations were estimated using the PLS-SEM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Our findings indicated that pre-use emotions did not significantly influence perceived personalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' However, satisfaction with personalisation, and perceived usefulness mediated by satisfaction, increased reuse intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' In addition, post-use positive emotions potentially influenced reuse intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Our paper, therefore, illustrates the applicability of theory regarding continuance use intentions to cancer-support websites and highlights the importance of personalisation for these purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Keywords: cancer website;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' continuance intention;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' emotions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' perceived usefulness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' satisfaction with personalization 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Introduction The demand for digital health has been rising 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' However, challenges and barriers to its usage have also become more evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' A number of prior studies, thus, advocated for personalisation 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', tailoring content to user needs, to improve engagement with digital health systems 2, and increase user satisfaction and usage intentions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Online health service personalisation has been explored from the perspective of: its association with self-disclosure 5, systematic literature reviews which have highlighted its benefits 6, personalised educational health content for the elderly 7 and adolescents 8, distress-based personalised therapy recommendations 9, and recommending health and fitness content for runners 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Nevertheless, research exploring the different factors that might influence intentions to (re)use personalised online health services is lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Our study addresses this gap, specifically for cancer-support websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Moreover, much has been discovered about how information technology (IT) use intentions 2 are affected by cognitive factors, as are perceived usefulness and satisfaction 11-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' However, there is insufficient understanding of how a user’s affective states might influence their use of online personalisation 16, 17, although it is known that emotions impact human behaviour and perception 18, and are induced by different interactions 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Therefore, there is good reason to explore if emotions can lead to and/or be influenced by the use of personalised online health services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Indeed, persistent sadness or anxiety were shown to increase the likelihood of online health information seeking 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Some studies examined the effects of website design 21, interface aesthetics 22 and website features 23 on affective responses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', arousal and irritation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Furthermore, the possible links between emotions and online personalisation have been considered in the domains of: e-commerce 4, group decision support systems 24, e-learning 17 and emotion-aware recommender systems 9, 25, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' However, to the best of our knowledge, our study is the first to propose a research model that explores emotions and perception of personalisation as factors influencing continuance use intentions in the context of cancer-support websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Theoretical background and research model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Underlying theories The underlying theories for our conceptual framework were: i) the two-stage model of cognition change toward IT usage 27 (Figure 1, part 1), ii) affect theory (Figure 1, part 2) , which defines emotions as drivers of human behaviour 18, iii) and appraisal theories (Figure 1, part 3), which describe emotions as reactions 19 to the current context’s assessment 28, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' The two-stage model has previously been applied to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', digital learning technologies 15, however, not to personalised cancer-support websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' The model measures changes in beliefs and attitude from pre- to post-usage stages, and satisfaction as a post- usage affective state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Our conceptual framework integrates the belief- and affect-based constructs at different IT use stages, however, we only examine changes in affective state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Furthermore, we measure belief only at the post-use stage, specifically as the perceived usefulness of personalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' In the two-stage model, the satisfaction construct captures the extent of user satisfaction, pleasantness, content, and delight toward IT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' We, on the other hand, use this construct to represent post-use cognitive-based appraisal of different aspects of personalisation (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='4 and Table A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Conceptual framework - expanded two-stage model of IT usage intentions with emotions Importantly, the two-stage model provides only a limited understanding of the relationship between emotions and perceptions or behavioural intentions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', via satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Pre- and post-usage attitudes measure user perception (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', cognitive appraisal) of whether system use was good, wise, positive and effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' In contrast, our pre- and post-use emotions’ factors explicitly capture user emotions, which are a very specific affect type (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Hence, our framework extends the two-stage model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' It incorporates emotion-specific constructs, which were derived from affect theories 18, 19 and measured by user self- assessment of basic emotions’ intensities experienced at a particular moment of system use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Research model and hypotheses Building upon the conceptual framework from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1, Figure 2 illustrates our research model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' This section defines its constituting factors and hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Research model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Pre-use stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='During-/post-use stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Affective state ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Cognitive appraisal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Affective appraisal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Positive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Positive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Perceived ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Reuse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='emotions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='emotions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='usefulness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='intentions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Negative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Satisfaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Negative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='emotions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='emotions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Two-stage model of IT usage intentions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Affect theory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Affect theory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Appraisal theoryPre-use stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='During-/ post-use stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Satisfaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Pre-use ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='personalisation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='positive emotions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Usefulness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Reuse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='intentions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Pre-use ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Post-use ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='personalisation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='negative emotions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='positive emotions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='Post-use ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='negative emotions4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Reuse intentions Behavioural intention is an intent to perform certain behaviour 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Theories of reasoned action and planned behaviour introduced this concept, followed by its adoption in IT use 30 and continuance intentions 27 frameworks, and applications in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', online personalisation 4, 31 and online health services 1, 32, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Our study measures users’ intentions to revisit and use a personalised cancer-support website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Emotions (pre- and post-use) Emotions are high intensity, brief affective states 24 that begin quickly 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' We measured 12 emotions, selected based on the results from two prior studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' The first study 35 explored the likelihood that interest, fear, sadness, surprise, awe, anger, embarrassment, guilt, enjoyment, shame, happiness, contempt, or disgust stimulate online cancer information searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' The second study 36 examined the association between online personalisation needs, usage intentions, and 13 basic and possible basic emotions, as classified by 34, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' These two studies showed that only certain basic emotions play a significant role in online health information use or perceived personalisation needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' These were fear, sadness, awe, excitement, interest and surprise, and were hence all accounted for in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' We also added six other emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Anxiety, boredom and calmness (or neutral state) were included due to their frequent use in the related human-computer interaction (HCI) research 16, 38-41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Embarrassment, guilt and happiness were basic emotions that were re-evaluated in this study to balance the number of positive and negative emotions measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Happiness (or alternatively joy, enjoyment, pleasantness) is also one of the essential positive valence emotions often studied in HCI 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Based on vulnerability research 43, embarrassment and guilt were included to represent the cancer-affected people’s potential negative perception of their own self, state or actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Given that positive and negative emotions influence behaviour differently 18, we used previous research 29, 34, 44-47 to classify the 12 emotions into positive (denoted above in italics) and negative valence categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Furthermore, we measured emotions at two stages (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1): pre-use emotions at the beginning of website use, and post-use emotions after website use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Emotions as stimuli of perception or action The effect of emotions on IT perception has been addressed in a few studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' For example, emotional attachment influenced the perceived usefulness and attitude towards Facebook 48, and affective quality of smart watches was associated with their perceived usefulness 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' However, such research in the online health domain is very limited, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', indicating that 5 interest and excitement increased the perceived needs for personalisation 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Given the under-researched association between emotions and perceived personalisation, we hypothesised that: H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1: Pre-use positive emotions increase the perceived usefulness of personalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2: Pre-use negative emotions decrease the perceived usefulness of personalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Interestingly, there is more research on the association between emotions and behavioural intentions towards IT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Enjoyment was found to influence web use 41, anxiety influenced continuance use intention for mobile-health among elderly 49, emotions stimulated online search for cancer information 35, and interest increased cancer patients’ use of electronic health records 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Strong positive emotions, or absence of negative emotions, mediated the effect of personalisation on online purchase intentions 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' However, we found only one prior study 36 that used linear regression and showed a positive influence of interest on reuse intentions of partially personalised online cancer services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Due to the lack of research on cause-effect relations (or structural equation modelling) between emotions and user intentions on personalised cancer-support websites we hypothesised that: H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1: Post-use positive emotions increase reuse intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2: Post-use negative emotions decrease reuse intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Emotions as reactions or affective appraisal We argue that users’ appraisal of cancer-support website personalisation can evoke post- use positive and negative emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' This is based on research showing that certain website features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', colour, images, shapes) induced emotions 52, perceived usefulness of educational blogs increased liking and pleasantness 11, pedagogical agent’s adaptation intensified enjoyment and decreased boredom 42, personalisation predicted post-use positive emotions in online shopping 51, and online health information overload (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', lacking personalisation) influenced negative emotions 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' However, there is insufficient understanding about the impact of personalised online health services 52 on affective states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' We therefore postulated the following hypotheses: H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1: Perceived usefulness of personalisation positively influences post-use positive emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2: Perceived usefulness of personalisation negatively influences post-use negative emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Moreover, consistent with research that showed that satisfaction positively influenced attitude after using IT 27, 54, we hypothesised that: 6 H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1: Satisfaction with personalisation positively influences post-use positive emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2: Satisfaction with personalisation negative influences post-use negative emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Usefulness of personalisation Perceived usefulness relates to expectations about performance improvements as a result of using a service or product 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Our usefulness of personalisation factor adapted Davis’s 55 ‘perceived usefulness’ to evaluate the individual personalisation features implemented on the studied cancer-support website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' This approach, has also been applied to e-commerce 44 and personalised e-learning 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Previous work has shown that perceived usefulness had a significant positive impact on satisfaction in the use of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', online- and m-banking 57, 58, e-government 54, and digital textbooks 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Moreover, personalisation applied to online news 59, e-commerce 60 and online-banking 61 increased user satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Therefore, for cancer-support websites, we hypothesised that: H5: Perceived usefulness of personalisation increases satisfaction with personalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Furthermore, previous research has argued that perceived usefulness 55 is an essential criterion for system reuse 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' It increased usage intentions for digital textbooks 15, health- information portals 32 and mobile health applications 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Hence, we hypothesised that: H6: Perceived usefulness of personalisation increases reuse intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Satisfaction with personalisation In a seminal work on customer expectations, Oliver 64 defines satisfaction as the “summary psychological state resulting when the emotion surrounding disconfirmed expectations is coupled with the consumer’s prior feelings about the consumption experience”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Abundant research shows that satisfaction positively affects online repurchase intentions 65 and (continuance) usage intentions for online-banking 57, e-government 54, educational blogs 11, m-health 33, and personalised information reuse 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Accordingly, with respect to personalised cancer-support websites, we hypothesised that: H7: Satisfaction with personalisation increases reuse intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Study design and data collection We sampled people directly and indirectly affected by cancer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' : former/current patients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' caregivers – family and friends;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' and those preventatively seeking cancer information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' This was achieved by using: purposive sampling for cancer association members;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' and convenience sampling for university students (as primary users of online health services 66), social networks’ users, and crowdsourced participants (via Amazon Mechanical Turk1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' This study was reviewed and approved (REGO-2015-1421) by the Biomedical and Scientific Research Ethics Committee at the University of Warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' It used an online survey methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' The survey first explained the study’s objective, the right to withdraw without consequences, and informed participants they were consenting to take part in this research and the collection of their anonymised responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' The following were then presented in a consecutive order: (1) Questionnaire: Reporting pre-use emotions (≈ 5 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' (2) Experiment: User-website interaction (≈ 25 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' (3) Questionnaire: Post-use emotions and website evaluation (≈ 30 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' PORT cancer website 1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='mturk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='com/ PORT WELCOME,SUNGM MYPORT NGO helping cancer patients in Bosnia and Herzegovina ABOUTUS ARTICLES KNOWLEDGEBASE VIRTUALCOMMUNITY Emotiontool MYPROFILE BLOG READLIST USERSFOLLOWED ACTIVITIES emotions,andtowhichextent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' ★WHATDIDYOU THINK ABOUTTHEFOLLOWING CONTENT?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=" Fear Immunotherapy-'Cureforterminal cancer found in game- Interest changing drugs Sadness Surprise Name:sungm Avwe RECOMMENDEDCONTENT EDIT Happiness Embarrassment OCHATROOM Excitement 3/5 KNOWLEDGEBASE Guit 21/03/1614:37 Cancerassociationsin Bosniaand Herzegovina Anxiety Is this a use Boredom Calmness KNOWLEDGE BA SE 04/03/1622:05 Why does cancer reoccur?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='8 During the experiment (step 2), participants interacted with PORT 67, a personalised cancer- support website (Figure 3) for patients and caregivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' PORT’s cancer-related content included cancer patients’ blogs, and articles adopted from respectable online sources about different cancer types, treatments and therapies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Participants completed the following2: user-profile creation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' privacy policy customisation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' user-profile editing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' interface adaptation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', adjusting font, colour, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' rating content and reviewing recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Since we were interested in the effect of interaction with a personalised cancer-support website, these tasks were essential for a user to explore the website, receive and perceive the personalisation, which on PORT comprised cancer information recommendations and user interface adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' The questionnaires (steps 1 and 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' see Appendix A) collected data on pre-use emotions, user demographics, post-use emotions, perceived usefulness of and satisfaction with personalisation, and website reuse intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' The scale for measuring emotion intensity was adopted from a game experience questionnaire 68, applied to online systems 44, 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Items from validated instruments were used for satisfaction with personalisation 70, 71 and reuse intentions 51, 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' The perceived usefulness instrument 55 was modified to measure the usefulness of individual personalisation features implemented on the PORT website (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' The online survey started in May 2015, and ran for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='5 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' We received 122 responses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 98 were valid and used in data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' We removed the data from respondents who were not affected at all or not interested in cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Data analysis and instrument validation Data pre-processing, exploratory factor analysis (EFA) and descriptive analyses were conducted using IBM SPSS® Statistics3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' SmartPLS 34 was used for confirmatory factor analysis (CFA) and structural equation modelling (SEM) with partial least squares (PLS) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' EFA was only applied to the 24 items for usefulness of personalisation (Appendix A), as we modified the original instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' We used principal axis factoring 73, direct Oblimin rotation 74, with Kaiser normalisation and a fixed number of factors based on our previous studies, which were confirmed with Eigenvalues>1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='0 and a scree test 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' A two-factor solution was selected: 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='89% variance explained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Eigenvalues>1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='43;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' KMO = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='76;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' χ2(55) = 440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='03, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 7 items reflected the factor usefulness of content-related personalisation (UsfCP), and 4 items represented the factor usefulness of explicit UI- and 2 See Appendix A for the complete list of website features participants were exposed to, asked to interact with and evaluate on perceived usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 3 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='com/uk-en/products/spss-statistics 4 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='smartpls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='com/ 9 content-adaptation (UsfADP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Namely, UsfCP covers the automatically generated recommendations of different content (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', articles, blog posts) and the content rating functionality required for these purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' UsfADP, on the other hand, refers to website features requiring more explicit user involvement for content customisation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', notifications and privacy policy length) and text appearance adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' We next ran a CFA in SmartPLS on the refined model with eight factors: pre-use positive (PREPE) and negative emotions (PRENE), post-use positive (POPE) and negative emotions (PONE), usefulness of content-related personalisation (UsfCP), usefulness of UI- /content-adaptation (UsfADP), satisfaction (SAT) and reuse intentions (RI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Cronbach’s α and composite reliability ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 75 and AVE - average variance extracted >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='5 76 were achieved by iteratively removing items with low outer loadings - starting with <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='5, up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 73, 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Table 1 presents reliability and validity results, and Table A1 (Appendix A) factor loadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' The Fornell-Larcker criterion for discriminant validity was satisfactory 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Heterotrait-monotrait ratio 79 was <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='85 for all factors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' apart from pre-use to post-use negative emotions (HTMT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='907), likely due to the same constituting emotions: fear, sadness and guilt/embarrassment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' However, the latter was acceptable at the HTMTinference criterion 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Construct reliability and validity Factor Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' of items Mean (SD);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=" N Cronbach's α Composite Reliability AVE PREPE 2 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='731 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='581 PRENE 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='771 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='771 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='529 UsfCP 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='828 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='546 UsfADP 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='757 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='507 SAT 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='796 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='566 POPE 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='580 PONE 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='520 RI 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='820 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='820 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='604 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=" Participant demographics The respondents' age ranged from 18 to 57 years (Mean=27, SD=8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' The majority were from B&H (51%) and USA (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7%), and 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2% were female.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' They were mainly caregivers to a family member who suffered from cancer (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1%), preventatively sought cancer information (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='6%), had a friend suffering from cancer (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='3%), or were a cancer patient (1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' PLS-SEM results Model fit was tested with a consistent PLS algorithm - all LVs connected for initial calculation, 300 iterations, path weighting scheme, missing values replaced with a mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' SRMR (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='069 *saturated, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='181 **estimated model) met the recommended value of <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='08 80, while NFI (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='717*, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='587**) was slightly below the recommended 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='9-1 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Figure 4 shows the path coefficients (ß) and coefficients of determination (R2) after applying complete bootstrapping with 2000 subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Estimated model - path coefficients and significance levels The findings showed that four path coefficients were significant at p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='05, supporting H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1, H5, H7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' At the pre-use stage, negative emotions (specifically fear, guilt and sadness categories) decreased the usefulness of adaptation-related personalisation, however at p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1 (H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2: ß = -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='19, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='71, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='088).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' During website use, perceived usefulness of content personalisation (H5 - UsfCP: ß = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='32, t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='93, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='003) and adaptation (H5 - UsfADP: ß = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='49, t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='77, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='000) significantly increased satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' However, without a direct effect on post-use emotions or reuse intentions (H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1, H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2, H6 were not supported).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' At the during- and post-use stage, satisfaction with personalisation intensified positive emotions (H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1: ß = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='44, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Satisfaction (H7: ß = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='45, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='6, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='000), and potentially post-use positive emotions (H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1: ß = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='17, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='090), increased reuse intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Interestingly, post-use negative emotions did not influence and were not influenced by the factors in our model (H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2 and H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2 not supported).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' We also tested mediating effects (Table 2), based on Zhao’s method 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Post-use emotions were not significant mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Nevertheless, satisfaction fully mediated the effect of usefulness of content- and adaptation-related personalisation (UsfCP and UsfADP, respectively) on post-use positive emotions and reuse intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Pre-use stage I During-/ post-use stage Satisfaction with personalisation H7 H5 (R2=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='486) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='45** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='32** H5 H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1 H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1 Pre-use UsfCP H6 Reuse 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='21 (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='49** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='44** positive emotions Usefulness of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='001 (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' intentions H1:1 personalisation (R2=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='426) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='08 (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1 H6 H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1 (R2=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='036) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='05 (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='13 (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='17# H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='11 (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=') Post-use Pre-use UsfADP - H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1 positive emotions Usefulness of negative emotions 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='23 (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=') H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2 (R2=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='110) personalisation H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2 /H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='19# H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2 I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='14 (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=') (R2=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='054) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='21 (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='12 (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=') Dashed line / n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' : not significant H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='2 #p<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='23 (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=') p<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='05 Post-use ** p<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='01 negative emotions (R2=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='061)11 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Mediating effects IV M DV P1: IV->M P2: M->DV P3: IV->DV P1⋅P2 Result UsfCP SAT POPE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='32** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='44** n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='14# Full mediation UsfCP SAT RI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='32** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='45** n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='14* Full mediation UsfADP SAT POPE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='49** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='44** n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='22** Full mediation UsfADP SAT RI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='49** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='45** n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='22** Full mediation IV: independent variable, M: mediator, DV: dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' # p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' *p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' **p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Discussion Our findings imply that the essential factor explaining reuse intentions for cancer-support websites is satisfaction with personalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' It mediates the effect of usefulness of personalisation, and directly increases reuse intentions, as seen in numerous studies on continuance use intentions in other domains 11, 15, 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' We next discuss and generalise the key results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' First, pre-use emotions do not significantly affect perceived personalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Although prior online-health research indicated a possible effect of positive emotions on personalisation needs 36, our study showed that surprise and awe (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', positive-valence, high-arousal emotions) do not influence usefulness of personalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Furthermore, we found a marginally significant effect of negative emotions, such that fear, guilt and sadness jointly decrease the usefulness of explicit UI- and content-adaptation (UsfADP), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', the type of personalisation which requires explicit user involvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' This likely occurs because people in negative affective states are biased towards negative events/occurrences 83, hence might not perceive the benefits of personalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Overall, these are valuable findings, providing an insight into the online cancer-support context, and inviting exploration of alternative emotion taxonomies and their association with perceived personalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Second, contrary to our prediction, usefulness of personalisation does not directly impact post-use emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' However, usefulness of personalisation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', both content-related personalisation and explicit UI- and content-adaptation) intensifies post-use positive emotions when mediated by satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Our results are consistent with e-commerce research regarding negative emotions 51, however, there, personalisation increased positive emotions 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' This difference could stem from the different measurement methods: we observed discrete emotions, while Pappas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 51 measured positive or negative mood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' we evaluated the perceived usefulness of individual personalisation features, and they examined users’ willingness to be provided with personalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 12 Third, cognitive perception of personalisation is more important than its affective appraisal 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Almost 50% of variation in satisfaction with personalisation is explained by the personalisation’s perceived usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Thus, our findings align with the positive effect found in online banking 57, 58, e-government 54 and digital textbooks 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' While research has addressed the effect of satisfaction on attitude 27, 54, our study was the first to explore its influence on post-use emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Specifically, we found that satisfaction with personalisation intensifies post-use positive emotions, indicating a pleasant surprise after confirming positive or disconfirming negative expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Finally, contrary to the findings of prior research in other domains, reuse intentions for personalised cancer-support websites are not significantly explained by post-use negative emotions or perceived usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Post-use negative emotions and benefits of personalisation affected online purchase intentions in 51, and negative affects, depressive symptoms and trait anger reduced online health information search intentions in 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Thus, behavioural intentions are possibly context- or task-dependent, or influenced differently by various affective states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' In fact, our findings suggest that post-use surprise and awe could increase cancer website reuse intentions, which aligns with the findings for positive emotions in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', online purchasing 31, 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Conclusion From a theoretical perspective, our research implies that the two-stage model’s constructs - usefulness and satisfaction - were applicable to understanding continuance use intentions for personalised cancer-support websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' However, alternative theories, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', Theory of Constructed Emotion 84, should be used for investigating the cause-effect between emotions and personalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Unlike the theory-proposed effect 18, emotions in the cancer-support website context were not a significant predictor of perceived personalisation or behavioural intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Nevertheless, we confirmed that context appraisal 28, 29, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=', perceived personalisation did evoke emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Furthermore, the frequently reported: i) effect of perceived usefulness on satisfaction with IT, and ii) the influence of satisfaction with IT on its reuse intentions, also prevail in cancer-support websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Our study’s important contribution was measuring perceived usefulness and satisfaction in relation to personalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Furthermore, this paper offers practical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Cancer-support website providers should implement personalisation, particularly content recommendations and interface adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' These features increase satisfaction and positive emotions, hence stimulate website reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 13 Our research, however, has limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Although comparable to computer-use studies 17, 85, 86, our sample size was relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' The sampled participants here were mainly people indirectly affected by cancer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' future research should focus on cancer patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Our findings’ generalisability is currently limited to cancer-support websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Moreover, alternative emotion taxonomies could be examined and longitudinal studies for a deeper insight into perceptions of personalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' In conclusion, this research uniquely applied affect and IT usage theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Finally, its main contribution is highlighting the importance of the understudied factors – emotions and personalisation - in forming user intentions toward online cancer-related services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Appendix A Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Overview of questionnaire items, measurement scales, factors and factor loadings Factor Questionnaire items Outer loadings 5-point scale: 1: Not experiencing this emotion at all, 2: Mildly, 3: Moderately, 4: Very, 5: Experiencing this emotion extremely Pre-use positive emotions Awe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='807 Surprise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='715 Calmness, Excitement, Happiness, Interest (removed) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 Pre-use negative emotions Guilt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='738 Fear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='734 Sadness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='709 Anxiety, Boredom, Embarrassment (removed) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 Post-use positive emotions Surprise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='762 Awe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='761 Calmness, Excitement, Happiness, Interest (removed) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 Post-use negative emotions Embarrassment 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='814 Sadness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='692 Fear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='647 Anxiety, Boredom, Guilt (removed) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='64 5-point scale: 1: strongly disagree to 5: strongly agree Usefulness of … (I perceive as useful the personalisation feature…) …content-related personalisation (UsfCP) UsfCP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Recommendations for forum discussions 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='777 UsfCP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Recommendations for blogs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='750 UsfCP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Recommendations for articles/news 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='723 UsfCP4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Content rating 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='704 UsfCP5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Recommendations for knowledge-base content (removed) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 UsfCP6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Personal readlist (removed) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 14 UsfCP7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Categorising content (popularity, recency, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=') (removed) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 …explicit UI- and content- adaptation (UsfADP) UsfADP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Privacy policy customisation (long/concise) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='749 UsfADP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Notifications/reminders 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='727 UsfADP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Text size adaptation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='657 UsfADP4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Text colour adaptation (removed) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='65 …other evaluated personalisation features (removed after EFA) Tailoring background colour;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' User-profile customisation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Defining interests;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Feedback about personalisation usefulness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' “What did you think about this content?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Emotion tool;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Filtering search;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Recommendations matching user’s interests;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Recommendations based on ratings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Recommendations based on user similarity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Filtering recommendations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Customising language;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Greetings with username Satisfaction with personalisation (I am satisfied with how PORT’s website was personalised to my needs because it…) SAT1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' … provided content at the right level of detail 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='793 SAT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' … provided valuable content to me 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='751 SAT3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' … could save me time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='710 SAT4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' … knew what I wanted (removed) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 SAT5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' … took into consideration my interests and preferences to make recommendations to me (removed) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 SAT6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' … improved my search performance (removed) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 SAT7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' … provided relevant content to me (removed) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 SAT8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' … provided up-to-date content to me (removed) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 Reuse intentions RI1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Overall, I have a positive attitude toward using the website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='839 RI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Given the chance, I intend to use the website again 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='744 RI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' I would recommend the website to my friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='744 RI4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' I intend to use the website frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' (removed) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content='7 Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Declaration of conflicting interests None to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 15 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Hong Z, Deng Z and Zhang W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Examining factors affecting patients trust in online healthcare services in China: The moderating role of the purpose of use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' Health informatics journal 2018: 1460458218796660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQf9_rY/content/2301.00886v1.pdf'} +page_content=' 2.' metadata={'source': 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+arXiv:2301.13164v1 [math.NA] 9 Dec 2022 +A Rellich’s result revisited and sensitivity of +solutions of parametrized linear systems +Jos´e Carlos Bellidoa, Luis Felipe Prieto-Mart´ınezb +aDepartamento de Matem´aticas, ETSI Industriales, INEI, Universidad de Castilla-La Mancha, Campus +Universitario S/N, E13071, Ciudad Real, Spain +bDepartamento de Matem´atica Aplicada, ETS Arquitectura, Universidad Polit´ecnica de Madrid, Avd. +Juan de Herrera 4, E28040, Madrid, Spain +Abstract +In this paper we revisit a result due to Franz Rellich on smoothness of so- +lutions of parametrized linear systems. With this result as a starting point, +we obtain finer smoothness results in an elementary fashion and propose an +efficient adjoint algorithm for computing sensitivities of (n − 1)-deficient sys- +tems, being n the order of the system. +Keywords: +Smoothness of solutions of linear systems with respect to +parameters, Derivates of linear systems solutions with respect to paramaters +2010 MSC: 65K99, 49K40, 90C31 +1. Introduction +More than 50 years ago Franz Rellich pioneered the investigation on the +problem of perturbation of eigenvalue problems with respect to matrix system +parameters, both for finite and infinite dimensional systems [1]. In this paper +we focus our attention on the following simple lemma from [1] in the finite +dimensional framework, where smoothness with respect to a single parameter +of solutions of homogeneous finite dimensional linear systems verifying a +normalization constraint is established. +Theorem 1 (Rellich’s Lemma). Let D(ε) = [γij(ε)]1≤i,j≤n be a n × n matrix +whose coefficients, γij(ε), for i, j = 1, . . . , n, are real analytic functions in a neighbor- +hood of some ε0 and such that for each ε in this neighborhood, det(D(ε)) = 0. Then, +for some neighborhood of ε0, there exist analytic functions α1(ε), . . . , αn(ε) such that +the column vector x(ε) = [α1(ε), . . . αn(ε)]T satisfies: +a) D(ε)x(ε) = 0, and +Email addresses: josecarlos.bellido@uclm.es (Jos´e Carlos Bellido), +luisfelipe.prieto@upm.es (Luis Felipe Prieto-Mart´ınez) +Preprint submitted to +January 31, 2023 + +b) ∥x(ε)∥ = 1. +In this paper we revisit this result, whose proof is elementary, exploring +the generalization of this theorem to the multi-parameter case, i.e. when coef- +ficient matrix D depends on a vector of variables ε ∈ RN, for both for homo- +geneous and non-homogeneous linear systems. More concretely, we would +like to provide answers to the following questions. +Problem 2. Let D(ε) = [γij(ε)]1≤i,j≤n, for ε ∈ U ⊂ RN. If for 1 ≤ i, j ≤ n, γij(ε) +is analytic (respectively in Cl(U)), decide: +• Given b(ε) = [b1(ε), . . . , bn(ε)]T, with bi(ε) analytic (resp. in Cl(U)) for +1 ≤ i ≤ n, does the system +D(ε)x(ε) = b(ε), +(1) +admits analytic solutions? +• does the linear system +D(ε)x(ε) = 0 +(2) +admits a solution x(ε) = [x1(ε), . . . , xn(ε)]T satisfying +∥x(ε)∥ = 1. +(3) +and such that each xi(ε) is analytic (resp. is in C1(U))? +A second step in the direction set in Problem 2 would be to device an +efficient way of computing derivatives or differentials of solutions of linear +systems with respect to parameters. Thus, the second question we address in +this investigation is the following. +Problem 3. Let D(ε) = [γij(ε)]1≤i,j≤n, for ε ∈ U ⊂ RN. If, for 1 ≤ i, j ≤ n, +the coefficients γij(ε) are analytic (resp. are in C1(U)) for 1 ≤ m ≤ N, determine +the value of +∂x +∂εm (ε0), where x(ε) is one of the analytic solutions (resp. C1(U)) of (2) +satisfying (3), and such that x(ε0) = u (with u a fixed given unitary vector such that +D(ε)u = 0). +Both, studying smoothness of solutions and the effective computation of +derivatives, usually referred as sensitivities, fit into the field of sensitivity analy- +sis, which for this kind of linear problems goes back to the times of A. Turing, +who in his influential paper [2], pointed out the interest on the problem of +sensitivity of solutions of linear systems (in that paper, he also introduced the +definition of condition number). The literature on the subject itself, or on inti- +mately related topics, is really vast. Rellich’s result that motivates this paper +is contextualized as an auxiliary result in the infancy of the perturbation the- +ory of eigenproblems [1]. [3] is a classical reference on the subject. Sensitivity +2 + +of eigenvalues and eigenvectors of linear problems is a matter of great prac- +tical interest in many engineering contexts, as for instance and just for citing +one among many, structural design [4, 5]. Of course, the problems we study +could be seen as particular cases of eigenvector sensitivity analysis, although +our questions are even more elementary and our aim here is to check how far +we can get pushing forward the elementary ideas in the proof of Theorem 1. +Literature on this subject is really overwhelming both from the mathematical +and engineering sides, and to include here an exhausting list of references is +out of the scope of this paper. We additionally reference [6, 7]. +Another very related subject is singular value decomposition of parametrized +matrices, where both the smoothness of the decomposition and methods for +computing it are addressed. The literature on this topic is also huge. We cite +[8, 9, 10, 11]. +Coming back to our very elementary problem on smoothness and sensi- +tivity of solutions of parametrized linear systems, it is worth mentioning the +highly cited survey [12]. See also [13]. The standard approach in these works +is the following: starting from a system Dx = b, with D a non-singular ma- +trix, consider the perturbed system (D + ∆D)x = b + ∆b, where ∆D and ∆b +are interpreted as perturbations or errors. Thus, these problems are usually +studied from the Numerical Linear Algebra viewpoint and from what is called +interval analysis in the literature. This viewpoint is mainly oriented to comput- +ing bounds for the uncertainty of the components of x rather than to compute +derivatives. +On the contrary, we assume coefficients of D, b are smooth functions de- +pending on a vector of parameters ε ∈ RN, instead of considering pertur- +bations. +This is the natural framework in many applied situations, where +the coefficients of the matrix sometimes depend on some parameters with +some (physical, or economical, or engineering) meaning. Furthermore, in the +case of homogeneous systems the constraint (3) is taken into consideration. +Smoothness and explicit computation of so-called frame solutions, i.e. families +of orthonormal solutions of the homogeneous system, has been studied in- +tensively before [14, 15, 16]. In this paper, we will provide a simple proof for +the existence of smooth frame solutions and an efficient method for comput- +ing sensitivities in the multi-parameter case for (n − 1)-deficient systems (i.e. +rank(D(ε)) = n − 1). +The existence of smooth solutions to linear systems with respect to param- +eters has received some recent attention due to its relationship with Whitney’s +Extension Problem. We highlight the recent articles [17, 18, 19], for the case +U = RN, where a complete characterization of the linear systems admitting +such a solution is obtained. We are not so interested in obtaining such char- +acterizations, but simple and coarse criterions for the existence of smooth so- +lutions, following the philosophy of [1], and to device methods for sensitivity +computation. +Outline of the paper is the following. In Section 2 we include an updated +version of the proof of Theorem 1 and, after this, we discuss the questions +in Problem 2 as generalizations of Theorem 1: the necessity of the analytic- +3 + +ity condition (Subsection 2.1), the multi-parameter case (Subsection 2.2) and +finally the general non-homogeneous case (Subsection 2.3). In Section 3 we +study the existence of smooth frame solutions when rank (D(ε)) < n − 1. Fi- +nally, Section 4 is devoted to the exposition of two sensitivities computation +algorithms for the multi-parameter case and for (n − 1)-deficient systems. We +provide a direct method, inspired in Nelson’s Method for simple eigenvec- +tor derivative calculation [20], and an adjoint method (more efficient for large +values of N) inspired by [4, 5]. +2. Rellich’s Theorem +In this section, first, we include, for readers’ convenience and to under- +stand the new results in this paper, the proof of Theorem 1. In Subsections +2.1, 2..2 and 2.3 extensions and generalizations of Theorem 1 are given, par- +tially answering Problem 2. +Proof of Theorem 1: +Let us denote by U the neighborhood of ε0 where the +entries of matrix D(ε) are analytic and such that det(D(ε)) = 0 for all ε ∈ U. +Set r = maxε∈U(rank(D(ε))). We may assume that D(ε) is not the trivial +matrix in U. So 1 ≤ r ≤ n − 1. +First we construct an analytic solution verifying Theorem 1, a). There is +no loss in generality (performing a permutation of the equations and of the +variables) in assuming that det([γij(ε)]r +i,j=1) is a minor which is not constantly +equal to 0 in U. +For 1 ≤ i, j ≤ n, denote by Γij(ε) to the cofactor of γij(ε) in the submatrix +[γij(ε)]r+1 +i,j=1. Defining +fk(ε) = +� +Γr+1,k(ε) +for k = 1, . . . , r + 1 +0 +for k = r + 2, . . . , n, +(4) +functions fk(ε) are analytic and not simultaneously constantly zero (because +fr+1(ε) = Γr+1,r+1(ε) ̸= 0 for some ε ∈ U). Moreover, we have that for i = +1, . . . , n, +[γi1(ε), . . . , γin(ε)] + + +f1(ε) +... +fn(ε) + + = +r+1 +∑ +k=1 +γik(ε)Γr+1,k(ε) = 0, +since the second term in the equality is the determinant of the (r + 1) × (r + 1) +matrix obtained by replacing the (r + 1)-st row of [γij(ε)]r+1 +i,j=1 by [γi1(ε), . . . , γi,r+1(ε)] +and therefore vanishes: +• for 1 ≤ i < r, since it is the determinant of a matrix with two equal rows; +• for i = r, since it is the determinant of matrix [γij(ε)]r+1 +i,j=1; +• for r < i ≤ n, since it is the determinant of a matrix such that its last +row is linear combination of the rest of rows. +4 + +In order to complete the proof let us assume, for the shake of simplicity, +that ε0 = 0. There exist some m ∈ N ∪ {0} such that, for 1 ≤ i ≤ n, the power +series fi(ε) is of order at least m. So we have: +(| f1(ε)|2 + . . . + | fn(ε)|2) = ε2m(h0 + h1ε + . . .), +h0 ̸= 0 +(5) +where the power series h0 + h1ε + . . . converges for sufficiently small |ε|. Con- +sequently: +(| f1(ε)|2 + . . . + | fn(ε)|2) +1 +2 = εm(ω0 + ω1ε + . . .), +ω0 ̸= 0 +where the power series ω0 + ω1x + . . . = √h0 + h1x + . . . converges for small +|ε|. Since, as we have established, the power series expansion of each fi(ε) are +of order at least m, we can define +αi(ε) = +fi(ε) +εm(ω0 + ω1ε + . . .), +i = 1, . . . , n +which is also a convergent power series for small |ε| and, for some 1 ≤ i ≤ m, +αi(ε) ̸= 0. +✷ +In the second part of the proof, the following elementary property of the +set of formal power series has been essential. This property will be recalled +later. +Property 4. If f1(ε), . . . , fk(ε) are convergent power series in a neighborhood of ε0 ∈ +R, then there exists some m ∈ N ∪ {0}, such that α1(ε) = +f1(ε) +(ε−ε0)m , . . . , αk(ε) = +fk(ε) +(ε−ε0)m are all of them convergent power series and for some i, 1 ≤ i ≤ k, αi(ε0) ̸= 0. +In the following subsections, we address generalizations of Theorem 1 in +different directions. +2.1. Analyticity assumption in Rellich’s Theorem +As was already pointed out by Rellich, and due to the important role +played by Property 4 in the proof of Theorem 1, it is not possible to replace the +analyticity condition in Theorem 1 by a weaker smoothness condition. That +is, if the coefficients γij(ε) ∈ Cl(U), l ≥ 1, for 1 ≤ i, j ≤ n, in general it is not +possible to find α1(ε), . . . , αn(ε) ∈ Cl(U∗), with U∗ being a neighborhood of ε0 +and satisfying the conditions a) and b) in Theorem 1. The following example +is an adaptation of the one appearing in [1] (recall that in [1] the eigenproblem +is addressed) illustrating this fact. +5 + +Example 5. For n = 2, consider the following matrix, which entries are continuous +and have continuous derivatives of all orders in R: +D(ε) = + + + + + + + + + + + + + + + + + + +e− 1 +ε2 (1 − cos 2 +ε ) +−e− 1 +ε2 sin 2 +ε +−e− 1 +ε2 sin 2 +ε +e− 1 +ε2 (1 + cos 2 +ε ) + + +for ε ̸= 0 +� +0 +0 +0 +0 +� +for ε = 0 +For ε ̸= 0 and for some real function λ(ε), any solution takes the form: +x(ε) = λ(ε) · +� cos(1/ε) +sin(1/ε)) +� +There is no neighborhood U∗ of ε0 = 0 with a vector x(ε) = +� +α1(ε) +α2(ε) +� +such that +α1(ε), α2(ε) are continuous and such that for every ε ∈ U∗ they satisfy a) and b) in +Theorem 1. +What is possible is, starting from a matrix with Cl(U) coefficients, to ob- +tain a vector x(ε) which components are also in Cl(U) and such that satisfy +a) in Theorem 1 and different o zero in some neighborhood of ε0, but not +Condition b). Example 5 also illustrates this situation. Starting from a matrix +whose coefficients are C∞(R), there exist a column vector x(ε) (which is not, +in general, unique) satisfying a) and such that its components are also C∞(R). +Take, for instance, +x(ε) = e− 1 +ε2 · +� +cos(1/ε) +sin(1/ε)) +� +. +In other words, it is possible to obtain the following result in the spirit of +Theorem 1 for weaker smoothness conditions. +Theorem 6. Let ε0 ∈ R, and let U be a neighborhood of ε0. Let D(ε) = [γij(ε)]1≤i,j≤n +a matrix such that each γij(ε) ∈ Cl(U), for all i, j = 1, . . . , n, and such that +det(D(ε)) = 0 for all ε ∈ U. Then: +1. There exist functions f1(ε), . . . , fn(ε) in Cl(U) such that the column vector +x(ε) = + + +f1(ε) +... +fn(ε) + + +satisfies +D(ε)x(ε) = 0 +for any ε ∈ U; +6 + +2. Let r = maxε∈U(rank(D(ε))) and assume that rank(D(ε0)) = r. Then, +there exists a neighborhood U∗ of ε0 such that for ε ∈ U∗, x(ε) ̸= 0, and +consequently, there exist functions α1(ε), . . . , αn(ε) in Cl(U∗) such that the +column vector +x(ε) = + + +α1(ε) +... +αn(ε) + + +satisfies conditions a) and b) in Theorem 1, for any ε ∈ U∗. +Proof: +The proof follows the lines of that of Theorem 1. Assume, again, +that 0 < r < n. There is no loss in generality assuming, also, that det([γij(ε)]r +i,j=1) +is a minor which is non trivial in some neighborhood U∗ ⊂ U of ε0. The proof +of part (1) is the same as the one of Theorem 1 (with the obvious modifi- +cations) and will not be repeated here. For the proof of Part 2, recovering +the previous notation, we just notice that since rank(D(ε0)) = r, therefore +constant and maximal in the whole neighborhood U∗, +| f1(ε)|2 + . . . + | fn(ε)|2 +is a Cl(U∗) function, which does not vanish for ε ∈ U∗. So does +(| f1(ε)|2 + . . . + | fn(ε)|2)1/2 +and so, for 1 ≤ i ≤ n +αi(ε) = +fi(ε) +(| f1(ε)|2 + . . . + | fn(ε)|2)1/2 +is in Cl(U∗). +✷ +The solutions constructed in the proof fail to have smoothness at the points +ε0 such that rank(D(ε0)) is not maximal (is less than r). This is the reason of +the problems appearing in the smoothness of the eigenvectors when two or +more eigenvalues coalesce in the eigenproblem context. +2.2. Multi-parameter case +Now let us consider the several variables case of the second part of Prob- +lem 2, that is, now D(ε) = [γij(ε)]1≤i,j≤n denotes a n × n matrix such that each +entry γij depends on a vector of variables ε ∈ RN, N > 1, and solutions are +vectors x(ε). +The natural analogue of Theorem 1 for this case is false, as we can see in +the following example. Again, this is a consequence of the fact that analytic +functions in several variables do not satisfy an analogue of Property 4. +7 + +Example 7. For n = 2, N = 2, consider the following matrix, which entries depend +on a vector of variables ε = (ε1, ε2) and are analytic functions for every (ε1, ε2) ∈ R2: +D(ε1, ε2) = +� +2ε1ε2 +ε2 +2 − ε2 +1 +0 +0 +� +Any solution of system +D(ε1, ε1)x(ε1, ε2) = 0 +verifies +x(ε1, ε2) = λ(ε1, ε2) · +� +ε2 +1 − ε2 +2 +2ε1ε2 +� +, +for some real function λ(ε1, ε2). +Now if we impose x(ε1, ε2) to satisfy Equation (2), we obtain +∥[ε2 +1 − ε2 +2, 2ε1ε2]T∥ = +� +(ε2 +1 − ε2 +2)2 + (2ε1ε2)2 = ε2 +1 + ε2 +2 +so the solution x(ε1, ε2) such that ∥x(ε1, ε2)∥ = 1 for (ε1, ε2) ̸= (0, 0) is (up to sign) +x(ε1, ε2) = + + +ε2 +1−ε2 +2 +ε2 +1+ε2 +2 +2ε1ε2 +ε2 +1+ε2 +2 + + +It is not possible to extend such a solution to be continuous at (0, 0). +In a similar fashion to what happened in the previous section, for a matrix +D(ε) which entries are smooth (analytic or in Cl(U)), it is possible to find a +vector which entries are also smooth (analytic or in Cl(U)) and such that it +satisfies (2), but not (3), in general, that is, it is possible to prove an analogue +to Theorem 6 for several variables, but not of Theorem 1. +Theorem 8. Let D(ε) = [γij(ε)]1≤i,j≤n such that, for i, j = 1, . . . , n, γij(ε) is +analytic (resp. Cl(U)) in a neighborhood U of some ε0 ∈ RN and such that for each +ε ∈ U, det(D(ε)) = 0. Then: +1. There exist functions f1(ε), . . . , fn(ε) which are analytic (resp. Cl(U)) and +such that the column vector +x(ε) = + + +f1(ε) +... +fn(ε) + + +satisfies (2); +8 + +2. Let r = maxε∈U(rank(D(ε))). Assume that rank(D(ε0)) = r. Then, there +exists a neighborhood U∗ of ε0, such that there exists functions α1(ε), . . . , αn(ε) +which are analytic (resp. Cl(U∗)) and the column vector +x(ε) = + + +α1(ε) +... +αn(ε) + + +satisfies conditions (2) and (3) for ε ∈ U∗. +The proof follows completely the lines to the one of Theorem 6 and will +not be repeated. +2.3. Extension to non-homogeneous linear systems +Now we deal with the first question raised in Problem 2. In this case, +the corresponding result has a direct and elementary proof and unifies the +one-parameter and the multi-parameter cases: +Theorem 9. Let D(ε) = [γij(ε)]1≤i,j≤n, b = [b1(ε), . . . , bn(ε)]T such that their +entries are analytic (resp. Cl(U)) in a neighborhood U of some ε0 ∈ RN, N ≥ 1. +If rank(D(ε0)) = rank(D(ε0) | b(ε0)) = r and for every ε ∈ U rank(D(ε) | +b(ε)) ≤ r (⋆), there exists some neighborhood U∗ of ε0 such that there exist func- +tions g1(ε), . . . , gm(ε) which are analytic (resp. +Cl(U∗)) and such that x(ε) = +[g1(ε), . . . , gn(ε)]T satisfies (1) for ε ∈ U∗. +Proof: +There is no loss in generality assuming that det([γij(ε0)]1≤i,j≤r) is a +non-trivial minor. There is a neighborhood U∗ of ε0 contained in U such that +for every ε ∈ U∗, det([γij(ε0)]1≤i,j≤r) ̸= 0. Consider the matrix : +�D(ε) = + + +γ11(ε) +. . . +γ1r(ε) +γ1,r+1(ε) +. . . +γ1n(ε) +... +... +... +... +γr1(ε) +. . . +γrr(ε) +γr,r+1(ε) +. . . +γrn(ε) +0 +. . . +0 +... +... +In−r +0 +. . . +0 + + +where In−r denotes the identity matrix of size (n − r) × (n − r). +The unique solution of �D(ε)x(ε) = b(ε) is a solution of D(ε)x(ε) = b(ε). +So the result is a direct consequence of Cramer’s rule applied to the system +�D(ε)x(ε) = b(ε). +✷ +Condition (⋆) holds automatically if rank(D(ε)) is constant in U and equal +to n. +9 + +3. System of smooth linearly independent solutions to Problem 2 +Let us begin by introducing some notation. Let U ⊂ RN. For us a vector +field is a map x : U → Rn (indeed, this has already appeared above). We +say that it is analytic (resp. is Cl(U)) if so are its components. The following +definition clarifies the term frame field, which appears in the literature with +more than one meaning. +Definition 10. Let U ⊂ RN, a frame field is a map F : U → (Rn)k +ε = (ε1, . . . , εN) �−→ (x1(ε), . . . , xk(ε)) +such that, for every ε ∈ U, xi(ε) · xj(ε) = δij (so k ≤ n). Note that each xi is a +n-dimensional vector field in U. We say that it is analytic (resp. in Cl(U)) if so are +each vector field xi, for 1 ≤ i ≤ k. +3.1. Frame fields of solutions of homogeneous linear systems +In the context of the second case of Problem 2, in this subsection we prove, +in some cases, the existence of not only a vector field corresponding to a +solution, but a frame field (x1(ε), . . . , xk(ε)) consisting on k solutions. +The proccess used in this section generalizes an analogue construction +done, again, in [1] for eigenvalue problems. Let us start with the following +simple observation. +Lemma 11. Let D be a constant matrix of size n × n and rank r ≥ 1. Let k = n − r +and x1, . . . , xk be any orthonormal basis of solutions of the linear system Dx = 0. +Then the matrix +B = D + xkxT +k +of size n × n has rank r + 1 and satisfies +Bxi +� += 0 +for i = 1, . . . , k − 1 +̸= 0 +for i = k +Proof: +For i = 1, . . . , k − 1, taking into account that x1, . . . , xk are in the +kernel of D and constitute an orthogonal system we get +Bxi = (D + xkxT +k )xi = 0 +and analogously, having in mind, this time, that xkxk = 1 +Bxk = (D + xkxT +k )xk = xk ̸= 0 +Since Bx = 0 has at least k − 1 solutions, rank(B) ≤ r + 1. To see the +equality, we have to check that there is no more than k − 1 linearly indepen- +dent solutions of this system. Suppose that we have a vector x0 such that +{x0, . . . , xk} is orthonormal. Then +Bx0 = (D + xkxT +k )x0 = Dx0 +10 + +So, if Bx0 = 0 then Dx0 = 0 which leads to a contradiction with rank(D) = r. +✷ +This idea allow us to prove the following theorem, which is an improved +statement of Theorem 1 (in the one-parameter case). +This result does not +appear explicitely in [1]. +Theorem 12. Let U be a neighborhood of ε0 ∈ R. Let D(ε) = [γij(ε)]1≤i,j≤n be +such that each γij(ε), for 1 ≤ i, j ≤ n, is analytic in U and such that for each +ε ∈ U, det(D(ε)) = 0. Suppose that r = maxε∈U{rank(D(ε))} and let k = n − r. +Then there exist an analytic frame field (x1(ε), . . . , xk(ε)) such that for each ε in a +neighborhood of ε0, for 1 ≤ i ≤ k, D(ε)xi(ε) = 0. +Proof: +We proceed by induction. The case k = 1 is a trivial consequence +of Theorem 1. +For k > 1, using, again, this result we can obtain a vector that will be de- +noted by xk(ε) such that D(ε)xk(ε) = 0 and ∥xk(ε)∥ = 1. For each ε ∈ U con- +sider any orthonormal set of solutions {y1(ε), . . . , yk−1(ε), xk(ε)} of the system +D(ε)x = 0 containing the one above. Now define B(ε) = D(ε) + xk(ε)xT +k (ε). +B(ε) also satisfies the hypothesis of the theorem, with rank(B) ≤ r + 1. So +we can apply the induction hypothesis to obtain a frame field of k − 1 vectors +(x1(ε), . . . , xk−1(ε)) satisfying the requirements in the theorem. And since, for +each ε ∈ U they belong to the subspace spanned by y1(ε), . . . , yk−1(ε), the set +{x1(ε), . . . , xk−1(ε), xk(ε)} is an orthonormal set. +✷ +In a similar fashion, we can obtain a similar result replacing analyticity +by Cl(U) smoothness in the one-parameter case (N = 1) and in the multi- +parameter case (N > 1). +Theorem 13. Let ε0 ∈ RN, and let U be a neighborhood of this ε0. Let D(ε) = +[γij(ε)]1≤i,j≤n be such that each γij(ε), for 1 ≤ i, j ≤ n, is in Cl(U) and such that +for each ε ∈ U, det(D(ε)) = 0. Suppose that r = maxε∈U{rank(D(ε))} and let +k = n − r. Then there exist k vector fields x1(ε), . . ., xk(ε) which are Cl(U) such that +for each ε ∈ U, for each 1 ≤ i, j ≤ k, i ̸= j, xi(ε)Txj(ε) = 0 and D(ε)xi(ε) = 0. +Moreover, if rank(D(ε0)) = r, then there exists a neighborhood U∗ of ε0 such that, +for ε ∈ U∗, it is possible to choose these vectors in such a way that (x1(ε), . . . , xk(ε)) +is a frame field. +The proof follows the lines of the previous one. +Let us remark that, although Theorems 12 and 13 are stated as purely +existence result, the method exhibited in the proof of Theorem 12 is construc- +tive, as shown in the following example, that we hope it helps to clarify the +algorithm. Anyway, to compute the frame field using this method is not com- +putationally efficient, since it requires to compute too many determinants. +11 + +Example 14. In the case n = 3, N = 1, consider the matrix D(ε) = + + +ε +0 +0 +0 +0 +0 +0 +0 +0 + +. +For U = R, this matrix satisfies +rank(D(ε)) = +� +1 +if ε ̸= 0 +0 +if ε = 0 +so we expect to obtain a frame field of solutions consisting in 2 vector, defined in +R \ {0}. +The proof of Theorem 1 explains how to obtain one of the solutions, using certain +cofactors. This solution is +x1(ε) = +1 +∥[0, ε, 0]T∥ + + +0 +ε +0 + + = + + +0 +1 +0 + + . +Now, following the idea in Theorem 13 we repeat the same proccess, but for the matrix +B(ε) = D(ε) + x1(ε)x1(ε)T = + + +ε +0 +0 +0 +1 +0 +0 +0 +0 + + +obtaining the solution +x2(ε) = +1 +∥ +���� 0 +0 +1 +0 +��� , +��� ε +0 +0 +0 +��� , +��� ε +0 +0 +1 +��� +�T +∥ + + +���� +0 +0 +1 +0 +���� +���� +ε +0 +0 +0 +���� +���� +ε +0 +0 +1 +���� + + += + + +0 +0 +1 + + . +Note that, in this case, the frame field can be extended to the whole open set U, +although this may not happen in general. +Finally, note that this analytic frame field (x1(ε), . . . , xk(ε)) obtained with +this method, is not the only possible one satisfying the conditions. +Remark 15. Using Gram-Schmidt Method it is possible to find analytic (resp. in +Cl(U)) vector fields �xk+1(ε), . . . , �xn(ε) in such a way that +{x1(ε), . . . , xk(ε), �xk+1(ε), . . . , �xn(ε)} +(6) +is an orthonormal basis. +Let (y1(ε), . . . , yk(ε)) be a frame field. Then it satisfies the conditions if and only +if, there exists a matrix K(ε) that maps the elements in the basis (x1(ε), . . . , xk(ε)) +into elements in the basis (y1(ε), . . . , yk(ε)) of the form +K(ε) = P(ε) +� T(ε) +0 +0 +In−k +� +P(ε)−1 +(7) +12 + +where P(ε) is the matrix whose columns are the vectors in the base (6), T(ε) is a +matrix which entries are analytic (resp. are in Cl(U)) and such that for each ε ∈ U +belongs to the orthogonal group O(k) and finally In−k is the (n − k) × (n − k) +identity matrix. Note that the entries of yi(ε) are analytic (resp. are in Cl(U)). +A matrix whose entries depend on some variables and belongs to O(n) for each +value of these variables, such as K(ε), is sometimes called kinematic matrix. In +fact, we can view the frame field (y1(ε), . . . , yk(ε)) as the result of applying a rigid +motion to the original frame field (x1(ε), . . ., xk(ε)) in such a way that the entries +in the matrix of this rigid motion have the adequate smoothness with respect to the +variables in ε. +3.2. Linearly independent solutions for non-homogeneous linear systems +From Theorem 9 and the results in the previous subsection, it is easy to +prove the following: +Theorem 16. Let ε0 ∈ R, and let U be a neighborhood of this ε0. Let D(ε) = +[γij(ε)]1≤i,j≤n, b(ε) = (b1(ε), . . . , bn(ε))T be such that, for 1 ≤ i, j ≤ n, γij(ε), +bi(ε), are analytic for 1 ≤ i, j ≤ n. Suppose that r = rank(D(ε0)) = rank([D(ε0) | +b(ε0)]) and that for every ε ∈ U, rank([D(ε) | b(ε)]) ≤ r. Let k = n − r. Then there +exist a neighborhood U∗ of ε0, an analytic vector field xp(ε) and an analytic frame +field (x1(ε), . . . , xk(ε)) such that any analytic vector field x(ε) satisfying Equation +(1) can be writen as: +x(ε) = xp(ε) + λ1(ε)x1(ε) + . . . + λk(ε)xk(ε) +for some analytic functions λ1(ε), . . . , λn(ε). +Theorem 17. Let ε0 ∈ RN, for N ≥ 1, and let U be a neighborhood of this ε0. Let +D(ε) = [γij(ε)]1≤i,j≤n, b(ε) = (b1(ε), . . . , bn(ε))T be such that, for 1 ≤ i, j ≤ n, +γij(ε), bi(ε) are in Cl(U), for 1 ≤ i, j ≤ n. Suppose that r = rank(D(ε0)) = +rank([D(ε0) | b(ε0)]) and that for every ε ∈ U, rank([D(ε) | b(ε)]) ≤ r. Let +k = n − r. Then there exist a neighborhood U∗ of ε0, a Cl(U∗) vector field xp(ε) +and a Cl(U∗) frame field (x1(ε), . . . , xk(ε)) such that any Cl(U∗) vector field x(ε) +satisfying Equation (1) can be writen as: +x(ε) = xp(ε) + λ1(ε)x1(ε) + . . . + λk(ε)xk(ε) +for some functions λ1(ε), . . . , λn(ε) in Cl(U∗). +The existence of xp (the particular solution) is ensured by Theorem 9 and +the existence of the frame fields (homogeneous solutions) by Theorems 12 and +13. +13 + +4. Sensitivity Analysis +In this section we study Problem 3. The first two subsections deal with +the case in which rank(D(ε0)) = n − 1 and rank(D(ε)) ≤ n − 1 for ε in some +neighborhood of ε0. +In the first one, we present a direct method to solve +Problem 3. In the second one, an adjoint method is provided to perform this +same task. This second type of methods are, computationally, more efficient +for large values of N. Finally, in Subsection 4.3 we discuss the difficulties for +computation of sensitivities in the case rank(D(ε0)) < n − 1. +4.1. Direct Method +Let us begin with the following straightforward observation. Recall that +we are considering solutions verifying x(ε0) = u, for a given unitary vector u, +as in the formulation of Problem 3. +Lemma 18. Let ε0 ∈ RN and U be a open neighborhood of ε0. Let x(ε) be a C1(ε) +vector field. Suppose that x(ε) is a solution of the system D(ε)x(ε) = b(ε) for +ε ∈ U, where the entries in D(ε) and b(ε) are in C1(U). Then: +D(ε0) ∂x +∂εi +(ε0) = ∂b +∂εi +(ε0) − ∂D +∂εi +(ε0)u. +(8) +Proof: +Equation (8) is obtained taking derivatives from D(ε)x(ε) = b(ε). +✷ +Lemma 19. Let ε0 ∈ RN and let U be a open neighborhood of ε0. Let x(ε) be a +C1(ε) vector field. If ∥x(ε)∥ = 1 for ε ∈ U, then, for 1 ≤ i ≤ N: +uT ∂x +∂εi +(ε0) = 0. +(9) +Proof: +Equation (9) is obtained taking derivatives in x(ε)Tx(ε) = 1. +✷ +Combining Equations (8) and (9) we can obtain a way to compute the +derivatives inspired in Nelson’s method (that was originally developed for +eigenvector sensitivity, see [4, 5, 22]). Note that the general solution of Equa- +tion (8) must be of the form +∂x +∂εi +(ε0) = v + cu +(10) +for v being a particular solution of the system (8) and c a constant to be +determined. From the fact that uT ∂x +∂εi (ε0) = 0 we obtain that: +c = −vTu +(11) +So finally we get: +14 + +Direct Method for the homogeneous system +D(ε)x(ε) = 0 such that x(ε0) = u and rank(D(ε)) = n − 1 is a neighborhood of ε0. +1.- +We look for a particular solution v of the singular linear system D(ε0)v = − ∂D +∂εi (ε0)u. +2.- +Set c = −vTu. +3.- +Finally, ∂x +∂εi (ε0) = v + cu. +4.2. Adjoint Method +Suppose that, for some F : Rn → R, for some 1 ≤ j ≤ n, we want to +compute +∂ +∂εj (F(x(ε0))) where x(ε) satisfies Equations (2) and (3). Using this +function F we obtain some notational advantages, we study (effortlessly) a +more general problem and we can easily particularize this problem (taking +F(x(ε)) = xi) to recover ∂xi +∂εj (ε0) for i = 1, . . . , n. +For some p ∈ Rn, λ ∈ R that do not depend on the variables in ε, consider +the trivial equation: +F(x(ε)) = F(x(ε)) + pTD(ε)x(ε) + 1 +2λ(xT(ε)x(ε) − 1) +� +�� +� +=0 +Taking derivatives we get: +∂ +∂εj +(F(x(ε0))) = ∇F(x(ε0)) ∂x +∂εj +(ε0) + pT +� +∂D +∂εj +(ε0)x(ε0) + D(ε0) ∂x +∂εj +(ε0) +� ++ λ +� +xT(ε) ∂x +∂εj +(ε) +� +and re-organizing: +∂ +∂εj +(F(x(ε0))) = pT ∂D +∂εj +(ε0)x(ε0) + +� +∇F(x(ε0)) + pTD(ε0) + λxT(ε) +� +� +�� +� +(⋆⋆) +∂x +∂εj +(ε0) +To simplify the expression above, we can choose pT and λ in such a way that +(⋆⋆) = 0. To achieve this, we choose λ ∈ R to be the (unique) real number +such that the following linear system has a solution (at this point is where we +need that rank(D(ε)) = n − 1) and then we solve it to find p. +D(ε0)Tp = −λx(ε0) − ∇F(x(ε0))T +(12) +Putting these ideas together we obtain: +Adjoint Method for the homogeneous system +to compute partial derivatives of F(x(ε)) at ε0, where D(ε)x(ε) = 0 and x(ε0) = u. +1.- +Find λ, p such that: +D(ε0)Tp = −λu − ∇F(u)T. +2.- +∂ +∂εj (F(x(ε0))) = pT ∂D +∂εj (ε0)u. +15 + +Note that, p is not unique, that is, we can find two different solutions p1, p2 +satisfying (12) and so +(pT +1 − pT +2 )D = 0 +(13) +But the value of +∂ +∂εj (F(x(ε0))) does not vary. To check this, let us see that this +invariance is equivalent to: +(pT +1 − pT +2 )∂D +∂εj +x = 0 +and this equation is true as a consequence of Equations (8) and (13). +If we want to compute several partial derivatives +∂x +∂εj1 (ε0), . . . , ∂x +∂εjr (ε) the +first part of the method is common to all of them and the second one just +requires the computation of the corresponding derivative of D and a multipli- +cation. +4.3. Cases in which the solution of Problem 3 is not determined +If for all ε ∈ U, rank(D(ε)) = n − k, for k > 1, then Problem 3 cannot be +solved unless more information is provided. Let us see the following clarify- +ing example: +Example 20. For the case N = 1, n = 4, U = R and ε0 = 0. Consider the matrix +D(ε) = + + +ε +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + . +Consider some vector field x(ε) such that, for all ε ∈ R, +D(ε)x(ε) = 0, +x(0) = (0, 1, 0, 0)T +This information is not sufficient to determine x′(0). To check this, just see that for +every vector (0, 0, a, b)T in the linear space spanned by {(0, 0, 1, 0)T, (0, 0, 0, 1)T}, +the solution x(ε) = (0, cos ε, sin aε, sin bε) satisfies x′(ε0) = (0, 0, a, b). +In the case of eigenproblems this feature is well known. It corresponds to +the case in which λ(ε) is an eigenvalue of constant multiplicity h > 1. +Suppose that we have guaranteed the existence of a frame field (x1(ε), . . . , xk(ε)) +consisting of k vector fields which are solutions of (2) and that some orthonor- +mal vectors u1, . . . , uk are provided, so that it is prescribed that xj(ε0) = uj for +j = 1 . . . , k. In this case, the derivative of the solutions must satisfy some extra +conditions. For 1 ≤ j1, j2 ≤ k, j1 ̸= j2: +xj1(ε)xj2(ε) = 0 ⇒ uT +j1 +∂x +∂εi +(ε0) + uT +j2 +∂x +∂εi +(ε0) = 0 +16 + +Even with this extra conditions, the solution to Problem (3) is not determined. +It can be proved in a straightforward manner, studying the corresponding +linear system, that has (k +2) degrees of freedom, or just noting the following: +Remark 21. In the notation of Remark 15, see that for the frame field (x1(ε), . . . , xk(ε)) +obtained following the method in the proof of Theorems 12 and 13 any other frame field +(y1(ε), . . . , yk(ε)) satisfies the corresponding conditions if and only if +for 1 ≤ i ≤ j, yi(ε) = K(ε)xi(ε) +(14) +Now, if we impose that, for 1 ≤ i ≤ k, xi(ε0) = yi(ε0) = ui for some prescribed +vector ui, then K(ε0) = In and so K(ε) is in SO(n). Taking derivatives in Equation +(14) we have that +∂yi +∂εi +(ε0) = ∂K +∂εi +(ε0)ui + ∂xi +∂εi +(ε) +If we want K(ε) to be of the type explained in Equation (7) and to satisfy K(ε0) = In +we still have (k +2) free parameters since ∂T +∂εi (ε0) can be any skew-symmetric matrix of +size k × k (see [23]). +Ackwnoledgements +This work has been supported by the Spanish Agencia Estatal de Investi- +gaci´on through project PID2020-116207GB-I00 and Junta de Comunidades de +Castilla-La Mancha through project SBPLY/19/180501/000110. +References +References +[1] F. Rellich, Perturbation Theory of Eigenvalues Problems, Gordon and +Breach Science Publishers, New York-London-Paris, Berlin, 1969. +[2] A. M. Turing, Rounding-off errors in matrix processes, The Quarterly +Journal of Mechanics and Applied Mathematics 1 (1) (1948) 287–308. +[3] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1966. +[4] A. D. D. Ruiz, J. C. Bellido, Eigenvector sensitivity when tracking modes +with repeated eigenvalues, Computer Methods in Applied Mechanics +and Engineering 326 (2017) 338–357. +[5] G. H. Yoon, A. Donoso, J. C. Bellido, D. Ruiz, Highly efiicient general +method for sensitivity analysis of eigenvectors with repeated eigenvalues +withoug passing through adjacent eigenvectors, International Journal for +Numerical Methods in Engineering 121 (20) (2020) 4473–4492. +17 + +[6] A. +L. +Andrew, +K.-w. +E. +Chu, +P. +Lancaster, +Derivatives of eigenvalues and eigenvectors of matrix functions, SIAM J. +Matrix Anal. 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Fefferman, Continuous solutions of linear equations, From +Fourier analysis and number theory to Randon transforms and geom- +etry, Springer, New York (2013) 233–282. +[18] G. K. L. C. Fefferman, The brenner–hochster–koll´ar and whitney prob- +lems for vector-valued functions and jets, Revista Matem´atica Iberoamer- +icana 30 (3) (2014) 875–892. +18 + +[19] G. K. L. C. Fefferman, Solutions to a system of equations for cm functions, +Revista Matem´atica Iberoamericana 37 (3) (2020) 911–963. +[20] R. B. Nelson, Simplified calculation of eigenvector derivatives, AIAA +Journal 14 (9) (1976) 1201 – 1205, cited by: 800. doi:10.2514/3.7211. +URL https://www.scopus.com/inward/record.uri?eid=2-s2.0-0016993544&doi=10.2514%2f3.7211&partnerID=40&md5=e42b0a9d1ce187e2c8673cc6fc3edd2f +[21] T. Romanowicz, Sensitivity analysis of linear systems–a structural ap- +proach, International Journal of Systems Science 18 (1) (1987) 91–96. +[22] R. B. Nelson, Simplified calculation of eigenvector derivatives, AIAA +Journal 14 (9) (1976) 1201–1205. +[23] A. Baker, Matrix groups: An introduction to Lie group theory, Springer +Science & Business Media, 2012. +19 + diff --git a/bdFPT4oBgHgl3EQfwzUG/content/tmp_files/load_file.txt b/bdFPT4oBgHgl3EQfwzUG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0aa360e09b4cd6a8c3c159f72e91b481be921f3 --- /dev/null +++ b/bdFPT4oBgHgl3EQfwzUG/content/tmp_files/load_file.txt @@ -0,0 +1,715 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf,len=714 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='13164v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='NA] 9 Dec 2022 A Rellich’s result revisited and sensitivity of solutions of parametrized linear systems Jos´e Carlos Bellidoa, Luis Felipe Prieto-Mart´ınezb aDepartamento de Matem´aticas, ETSI Industriales, INEI, Universidad de Castilla-La Mancha, Campus Universitario S/N, E13071, Ciudad Real, Spain bDepartamento de Matem´atica Aplicada, ETS Arquitectura, Universidad Polit´ecnica de Madrid, Avd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Juan de Herrera 4, E28040, Madrid, Spain Abstract In this paper we revisit a result due to Franz Rellich on smoothness of so- lutions of parametrized linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' With this result as a starting point, we obtain finer smoothness results in an elementary fashion and propose an efficient adjoint algorithm for computing sensitivities of (n − 1)-deficient sys- tems, being n the order of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Keywords: Smoothness of solutions of linear systems with respect to parameters, Derivates of linear systems solutions with respect to paramaters 2010 MSC: 65K99, 49K40, 90C31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Introduction More than 50 years ago Franz Rellich pioneered the investigation on the problem of perturbation of eigenvalue problems with respect to matrix system parameters, both for finite and infinite dimensional systems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In this paper we focus our attention on the following simple lemma from [1] in the finite dimensional framework, where smoothness with respect to a single parameter of solutions of homogeneous finite dimensional linear systems verifying a normalization constraint is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Theorem 1 (Rellich’s Lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let D(ε) = [γij(ε)]1≤i,j≤n be a n × n matrix whose coefficients, γij(ε), for i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , n, are real analytic functions in a neighbor- hood of some ε0 and such that for each ε in this neighborhood, det(D(ε)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Then, for some neighborhood of ε0, there exist analytic functions α1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , αn(ε) such that the column vector x(ε) = [α1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' αn(ε)]T satisfies: a) D(ε)x(ε) = 0, and Email addresses: josecarlos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='bellido@uclm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='es (Jos´e Carlos Bellido), luisfelipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='prieto@upm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='es (Luis Felipe Prieto-Mart´ınez) Preprint submitted to January 31, 2023 b) ∥x(ε)∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In this paper we revisit this result, whose proof is elementary, exploring the generalization of this theorem to the multi-parameter case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' when coef- ficient matrix D depends on a vector of variables ε ∈ RN, for both for homo- geneous and non-homogeneous linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' More concretely, we would like to provide answers to the following questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let D(ε) = [γij(ε)]1≤i,j≤n, for ε ∈ U ⊂ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' If for 1 ≤ i, j ≤ n, γij(ε) is analytic (respectively in Cl(U)), decide: Given b(ε) = [b1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , bn(ε)]T, with bi(ε) analytic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' in Cl(U)) for 1 ≤ i ≤ n, does the system D(ε)x(ε) = b(ε), (1) admits analytic solutions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' does the linear system D(ε)x(ε) = 0 (2) admits a solution x(ε) = [x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xn(ε)]T satisfying ∥x(ε)∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' (3) and such that each xi(ε) is analytic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' is in C1(U))?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' A second step in the direction set in Problem 2 would be to device an efficient way of computing derivatives or differentials of solutions of linear systems with respect to parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Thus, the second question we address in this investigation is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let D(ε) = [γij(ε)]1≤i,j≤n, for ε ∈ U ⊂ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' If, for 1 ≤ i, j ≤ n, the coefficients γij(ε) are analytic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' are in C1(U)) for 1 ≤ m ≤ N, determine the value of ∂x ∂εm (ε0), where x(ε) is one of the analytic solutions (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' C1(U)) of (2) satisfying (3), and such that x(ε0) = u (with u a fixed given unitary vector such that D(ε)u = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Both, studying smoothness of solutions and the effective computation of derivatives, usually referred as sensitivities, fit into the field of sensitivity analy- sis, which for this kind of linear problems goes back to the times of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Turing, who in his influential paper [2], pointed out the interest on the problem of sensitivity of solutions of linear systems (in that paper, he also introduced the definition of condition number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The literature on the subject itself, or on inti- mately related topics, is really vast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Rellich’s result that motivates this paper is contextualized as an auxiliary result in the infancy of the perturbation the- ory of eigenproblems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' [3] is a classical reference on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Sensitivity 2 of eigenvalues and eigenvectors of linear problems is a matter of great prac- tical interest in many engineering contexts, as for instance and just for citing one among many, structural design [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Of course, the problems we study could be seen as particular cases of eigenvector sensitivity analysis, although our questions are even more elementary and our aim here is to check how far we can get pushing forward the elementary ideas in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Literature on this subject is really overwhelming both from the mathematical and engineering sides, and to include here an exhausting list of references is out of the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' We additionally reference [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Another very related subject is singular value decomposition of parametrized matrices, where both the smoothness of the decomposition and methods for computing it are addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The literature on this topic is also huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' We cite [8, 9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Coming back to our very elementary problem on smoothness and sensi- tivity of solutions of parametrized linear systems, it is worth mentioning the highly cited survey [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' See also [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The standard approach in these works is the following: starting from a system Dx = b, with D a non-singular ma- trix, consider the perturbed system (D + ∆D)x = b + ∆b, where ∆D and ∆b are interpreted as perturbations or errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Thus, these problems are usually studied from the Numerical Linear Algebra viewpoint and from what is called interval analysis in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' This viewpoint is mainly oriented to comput- ing bounds for the uncertainty of the components of x rather than to compute derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' On the contrary, we assume coefficients of D, b are smooth functions de- pending on a vector of parameters ε ∈ RN, instead of considering pertur- bations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' This is the natural framework in many applied situations, where the coefficients of the matrix sometimes depend on some parameters with some (physical, or economical, or engineering) meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Furthermore, in the case of homogeneous systems the constraint (3) is taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Smoothness and explicit computation of so-called frame solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' families of orthonormal solutions of the homogeneous system, has been studied in- tensively before [14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In this paper, we will provide a simple proof for the existence of smooth frame solutions and an efficient method for comput- ing sensitivities in the multi-parameter case for (n − 1)-deficient systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' rank(D(ε)) = n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The existence of smooth solutions to linear systems with respect to param- eters has received some recent attention due to its relationship with Whitney’s Extension Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' We highlight the recent articles [17, 18, 19], for the case U = RN, where a complete characterization of the linear systems admitting such a solution is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' We are not so interested in obtaining such char- acterizations, but simple and coarse criterions for the existence of smooth so- lutions, following the philosophy of [1], and to device methods for sensitivity computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Outline of the paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In Section 2 we include an updated version of the proof of Theorem 1 and, after this, we discuss the questions in Problem 2 as generalizations of Theorem 1: the necessity of the analytic- 3 ity condition (Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='1), the multi-parameter case (Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='2) and finally the general non-homogeneous case (Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In Section 3 we study the existence of smooth frame solutions when rank (D(ε)) < n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Fi- nally, Section 4 is devoted to the exposition of two sensitivities computation algorithms for the multi-parameter case and for (n − 1)-deficient systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' We provide a direct method, inspired in Nelson’s Method for simple eigenvec- tor derivative calculation [20], and an adjoint method (more efficient for large values of N) inspired by [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Rellich’s Theorem In this section, first, we include, for readers’ convenience and to under- stand the new results in this paper, the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In Subsections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='.2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='3 extensions and generalizations of Theorem 1 are given, par- tially answering Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Proof of Theorem 1: Let us denote by U the neighborhood of ε0 where the entries of matrix D(ε) are analytic and such that det(D(ε)) = 0 for all ε ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Set r = maxε∈U(rank(D(ε))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' We may assume that D(ε) is not the trivial matrix in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' So 1 ≤ r ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' First we construct an analytic solution verifying Theorem 1, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' There is no loss in generality (performing a permutation of the equations and of the variables) in assuming that det([γij(ε)]r i,j=1) is a minor which is not constantly equal to 0 in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' For 1 ≤ i, j ≤ n, denote by Γij(ε) to the cofactor of γij(ε) in the submatrix [γij(ε)]r+1 i,j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Defining fk(ε) = � Γr+1,k(ε) for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , r + 1 0 for k = r + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , n, (4) functions fk(ε) are analytic and not simultaneously constantly zero (because fr+1(ε) = Γr+1,r+1(ε) ̸= 0 for some ε ∈ U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Moreover, we have that for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , n, [γi1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , γin(ε)] \uf8ee \uf8ef\uf8f0 f1(ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' fn(ε) \uf8f9 \uf8fa\uf8fb = r+1 ∑ k=1 γik(ε)Γr+1,k(ε) = 0, since the second term in the equality is the determinant of the (r + 1) × (r + 1) matrix obtained by replacing the (r + 1)-st row of [γij(ε)]r+1 i,j=1 by [γi1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , γi,r+1(ε)] and therefore vanishes: for 1 ≤ i < r, since it is the determinant of a matrix with two equal rows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' for i = r, since it is the determinant of matrix [γij(ε)]r+1 i,j=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' for r < i ≤ n, since it is the determinant of a matrix such that its last row is linear combination of the rest of rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 4 In order to complete the proof let us assume, for the shake of simplicity, that ε0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' There exist some m ∈ N ∪ {0} such that, for 1 ≤ i ≤ n, the power series fi(ε) is of order at least m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' So we have: (| f1(ε)|2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' + | fn(ε)|2) = ε2m(h0 + h1ε + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' ), h0 ̸= 0 (5) where the power series h0 + h1ε + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' converges for sufficiently small |ε|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Con- sequently: (| f1(ε)|2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' + | fn(ε)|2) 1 2 = εm(ω0 + ω1ε + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' ), ω0 ̸= 0 where the power series ω0 + ω1x + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' = √h0 + h1x + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' converges for small |ε|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Since, as we have established, the power series expansion of each fi(ε) are of order at least m, we can define αi(ε) = fi(ε) εm(ω0 + ω1ε + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' ), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , n which is also a convergent power series for small |ε| and, for some 1 ≤ i ≤ m, αi(ε) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' ✷ In the second part of the proof, the following elementary property of the set of formal power series has been essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' This property will be recalled later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Property 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' If f1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , fk(ε) are convergent power series in a neighborhood of ε0 ∈ R, then there exists some m ∈ N ∪ {0}, such that α1(ε) = f1(ε) (ε−ε0)m , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , αk(ε) = fk(ε) (ε−ε0)m are all of them convergent power series and for some i, 1 ≤ i ≤ k, αi(ε0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In the following subsections, we address generalizations of Theorem 1 in different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Analyticity assumption in Rellich’s Theorem As was already pointed out by Rellich, and due to the important role played by Property 4 in the proof of Theorem 1, it is not possible to replace the analyticity condition in Theorem 1 by a weaker smoothness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' That is, if the coefficients γij(ε) ∈ Cl(U), l ≥ 1, for 1 ≤ i, j ≤ n, in general it is not possible to find α1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , αn(ε) ∈ Cl(U∗), with U∗ being a neighborhood of ε0 and satisfying the conditions a) and b) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The following example is an adaptation of the one appearing in [1] (recall that in [1] the eigenproblem is addressed) illustrating this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 5 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' For n = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' consider the following matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' which entries are continuous and have continuous derivatives of all orders in R: D(ε) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 \uf8ee \uf8f0e− 1 ε2 (1 − cos 2 ε ) −e− 1 ε2 sin 2 ε −e− 1 ε2 sin 2 ε e− 1 ε2 (1 + cos 2 ε ) \uf8f9 \uf8fb for ε ̸= 0 � 0 0 0 0 � for ε = 0 For ε ̸= 0 and for some real function λ(ε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' any solution takes the form: x(ε) = λ(ε) · � cos(1/ε) sin(1/ε)) � There is no neighborhood U∗ of ε0 = 0 with a vector x(ε) = � α1(ε) α2(ε) � such that α1(ε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' α2(ε) are continuous and such that for every ε ∈ U∗ they satisfy a) and b) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' What is possible is, starting from a matrix with Cl(U) coefficients, to ob- tain a vector x(ε) which components are also in Cl(U) and such that satisfy a) in Theorem 1 and different o zero in some neighborhood of ε0, but not Condition b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Example 5 also illustrates this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Starting from a matrix whose coefficients are C∞(R), there exist a column vector x(ε) (which is not, in general, unique) satisfying a) and such that its components are also C∞(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Take, for instance, x(ε) = e− 1 ε2 · � cos(1/ε) sin(1/ε)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In other words, it is possible to obtain the following result in the spirit of Theorem 1 for weaker smoothness conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let ε0 ∈ R, and let U be a neighborhood of ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let D(ε) = [γij(ε)]1≤i,j≤n a matrix such that each γij(ε) ∈ Cl(U), for all i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , n, and such that det(D(ε)) = 0 for all ε ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' There exist functions f1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , fn(ε) in Cl(U) such that the column vector x(ε) = \uf8ee \uf8ef\uf8f0 f1(ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' fn(ε) \uf8f9 \uf8fa\uf8fb satisfies D(ε)x(ε) = 0 for any ε ∈ U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let r = maxε∈U(rank(D(ε))) and assume that rank(D(ε0)) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Then, there exists a neighborhood U∗ of ε0 such that for ε ∈ U∗, x(ε) ̸= 0, and consequently, there exist functions α1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , αn(ε) in Cl(U∗) such that the column vector x(ε) = \uf8ee \uf8ef\uf8f0 α1(ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' αn(ε) \uf8f9 \uf8fa\uf8fb satisfies conditions a) and b) in Theorem 1, for any ε ∈ U∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Proof: The proof follows the lines of that of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Assume, again, that 0 < r < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' There is no loss in generality assuming, also, that det([γij(ε)]r i,j=1) is a minor which is non trivial in some neighborhood U∗ ⊂ U of ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The proof of part (1) is the same as the one of Theorem 1 (with the obvious modifi- cations) and will not be repeated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' For the proof of Part 2, recovering the previous notation, we just notice that since rank(D(ε0)) = r, therefore constant and maximal in the whole neighborhood U∗, | f1(ε)|2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' + | fn(ε)|2 is a Cl(U∗) function, which does not vanish for ε ∈ U∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' So does (| f1(ε)|2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' + | fn(ε)|2)1/2 and so, for 1 ≤ i ≤ n αi(ε) = fi(ε) (| f1(ε)|2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' + | fn(ε)|2)1/2 is in Cl(U∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' ✷ The solutions constructed in the proof fail to have smoothness at the points ε0 such that rank(D(ε0)) is not maximal (is less than r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' This is the reason of the problems appearing in the smoothness of the eigenvectors when two or more eigenvalues coalesce in the eigenproblem context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Multi-parameter case Now let us consider the several variables case of the second part of Prob- lem 2, that is, now D(ε) = [γij(ε)]1≤i,j≤n denotes a n × n matrix such that each entry γij depends on a vector of variables ε ∈ RN, N > 1, and solutions are vectors x(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The natural analogue of Theorem 1 for this case is false, as we can see in the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Again, this is a consequence of the fact that analytic functions in several variables do not satisfy an analogue of Property 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 7 Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' For n = 2, N = 2, consider the following matrix, which entries depend on a vector of variables ε = (ε1, ε2) and are analytic functions for every (ε1, ε2) ∈ R2: D(ε1, ε2) = � 2ε1ε2 ε2 2 − ε2 1 0 0 � Any solution of system D(ε1, ε1)x(ε1, ε2) = 0 verifies x(ε1, ε2) = λ(ε1, ε2) · � ε2 1 − ε2 2 2ε1ε2 � , for some real function λ(ε1, ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Now if we impose x(ε1, ε2) to satisfy Equation (2), we obtain ∥[ε2 1 − ε2 2, 2ε1ε2]T∥ = � (ε2 1 − ε2 2)2 + (2ε1ε2)2 = ε2 1 + ε2 2 so the solution x(ε1, ε2) such that ∥x(ε1, ε2)∥ = 1 for (ε1, ε2) ̸= (0, 0) is (up to sign) x(ε1, ε2) = \uf8ee \uf8f0 ε2 1−ε2 2 ε2 1+ε2 2 2ε1ε2 ε2 1+ε2 2 \uf8f9 \uf8fb It is not possible to extend such a solution to be continuous at (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In a similar fashion to what happened in the previous section, for a matrix D(ε) which entries are smooth (analytic or in Cl(U)), it is possible to find a vector which entries are also smooth (analytic or in Cl(U)) and such that it satisfies (2), but not (3), in general, that is, it is possible to prove an analogue to Theorem 6 for several variables, but not of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let D(ε) = [γij(ε)]1≤i,j≤n such that, for i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , n, γij(ε) is analytic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Cl(U)) in a neighborhood U of some ε0 ∈ RN and such that for each ε ∈ U, det(D(ε)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' There exist functions f1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , fn(ε) which are analytic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Cl(U)) and such that the column vector x(ε) = \uf8ee \uf8ef\uf8f0 f1(ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' fn(ε) \uf8f9 \uf8fa\uf8fb satisfies (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let r = maxε∈U(rank(D(ε))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Assume that rank(D(ε0)) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Then, there exists a neighborhood U∗ of ε0, such that there exists functions α1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , αn(ε) which are analytic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Cl(U∗)) and the column vector x(ε) = \uf8ee \uf8ef\uf8f0 α1(ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' αn(ε) \uf8f9 \uf8fa\uf8fb satisfies conditions (2) and (3) for ε ∈ U∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The proof follows completely the lines to the one of Theorem 6 and will not be repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Extension to non-homogeneous linear systems Now we deal with the first question raised in Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In this case, the corresponding result has a direct and elementary proof and unifies the one-parameter and the multi-parameter cases: Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let D(ε) = [γij(ε)]1≤i,j≤n, b = [b1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , bn(ε)]T such that their entries are analytic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Cl(U)) in a neighborhood U of some ε0 ∈ RN, N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' If rank(D(ε0)) = rank(D(ε0) | b(ε0)) = r and for every ε ∈ U rank(D(ε) | b(ε)) ≤ r (⋆), there exists some neighborhood U∗ of ε0 such that there exist func- tions g1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , gm(ε) which are analytic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Cl(U∗)) and such that x(ε) = [g1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , gn(ε)]T satisfies (1) for ε ∈ U∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Proof: There is no loss in generality assuming that det([γij(ε0)]1≤i,j≤r) is a non-trivial minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' There is a neighborhood U∗ of ε0 contained in U such that for every ε ∈ U∗, det([γij(ε0)]1≤i,j≤r) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Consider the matrix : �D(ε) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 γ11(ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' γ1r(ε) γ1,r+1(ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' γ1n(ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' γr1(ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' γrr(ε) γr,r+1(ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' γrn(ε) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In−r 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb where In−r denotes the identity matrix of size (n − r) × (n − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The unique solution of �D(ε)x(ε) = b(ε) is a solution of D(ε)x(ε) = b(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' So the result is a direct consequence of Cramer’s rule applied to the system �D(ε)x(ε) = b(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' ✷ Condition (⋆) holds automatically if rank(D(ε)) is constant in U and equal to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' System of smooth linearly independent solutions to Problem 2 Let us begin by introducing some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let U ⊂ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' For us a vector field is a map x : U → Rn (indeed, this has already appeared above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' We say that it is analytic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' is Cl(U)) if so are its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The following definition clarifies the term frame field, which appears in the literature with more than one meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let U ⊂ RN, a frame field is a map F : U → (Rn)k ε = (ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , εN) �−→ (x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk(ε)) such that, for every ε ∈ U, xi(ε) · xj(ε) = δij (so k ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Note that each xi is a n-dimensional vector field in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' We say that it is analytic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' in Cl(U)) if so are each vector field xi, for 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Frame fields of solutions of homogeneous linear systems In the context of the second case of Problem 2, in this subsection we prove, in some cases, the existence of not only a vector field corresponding to a solution, but a frame field (x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk(ε)) consisting on k solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The proccess used in this section generalizes an analogue construction done, again, in [1] for eigenvalue problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let us start with the following simple observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let D be a constant matrix of size n × n and rank r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let k = n − r and x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk be any orthonormal basis of solutions of the linear system Dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Then the matrix B = D + xkxT k of size n × n has rank r + 1 and satisfies Bxi � = 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , k − 1 ̸= 0 for i = k Proof: For i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , k − 1, taking into account that x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk are in the kernel of D and constitute an orthogonal system we get Bxi = (D + xkxT k )xi = 0 and analogously, having in mind, this time, that xkxk = 1 Bxk = (D + xkxT k )xk = xk ̸= 0 Since Bx = 0 has at least k − 1 solutions, rank(B) ≤ r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' To see the equality, we have to check that there is no more than k − 1 linearly indepen- dent solutions of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Suppose that we have a vector x0 such that {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk} is orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Then Bx0 = (D + xkxT k )x0 = Dx0 10 So, if Bx0 = 0 then Dx0 = 0 which leads to a contradiction with rank(D) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' ✷ This idea allow us to prove the following theorem, which is an improved statement of Theorem 1 (in the one-parameter case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' This result does not appear explicitely in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let U be a neighborhood of ε0 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let D(ε) = [γij(ε)]1≤i,j≤n be such that each γij(ε), for 1 ≤ i, j ≤ n, is analytic in U and such that for each ε ∈ U, det(D(ε)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Suppose that r = maxε∈U{rank(D(ε))} and let k = n − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Then there exist an analytic frame field (x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk(ε)) such that for each ε in a neighborhood of ε0, for 1 ≤ i ≤ k, D(ε)xi(ε) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Proof: We proceed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The case k = 1 is a trivial consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' For k > 1, using, again, this result we can obtain a vector that will be de- noted by xk(ε) such that D(ε)xk(ε) = 0 and ∥xk(ε)∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' For each ε ∈ U con- sider any orthonormal set of solutions {y1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , yk−1(ε), xk(ε)} of the system D(ε)x = 0 containing the one above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Now define B(ε) = D(ε) + xk(ε)xT k (ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' B(ε) also satisfies the hypothesis of the theorem, with rank(B) ≤ r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' So we can apply the induction hypothesis to obtain a frame field of k − 1 vectors (x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk−1(ε)) satisfying the requirements in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' And since, for each ε ∈ U they belong to the subspace spanned by y1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , yk−1(ε), the set {x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk−1(ε), xk(ε)} is an orthonormal set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' ✷ In a similar fashion, we can obtain a similar result replacing analyticity by Cl(U) smoothness in the one-parameter case (N = 1) and in the multi- parameter case (N > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let ε0 ∈ RN, and let U be a neighborhood of this ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let D(ε) = [γij(ε)]1≤i,j≤n be such that each γij(ε), for 1 ≤ i, j ≤ n, is in Cl(U) and such that for each ε ∈ U, det(D(ε)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Suppose that r = maxε∈U{rank(D(ε))} and let k = n − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Then there exist k vector fields x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=', xk(ε) which are Cl(U) such that for each ε ∈ U, for each 1 ≤ i, j ≤ k, i ̸= j, xi(ε)Txj(ε) = 0 and D(ε)xi(ε) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Moreover, if rank(D(ε0)) = r, then there exists a neighborhood U∗ of ε0 such that, for ε ∈ U∗, it is possible to choose these vectors in such a way that (x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk(ε)) is a frame field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The proof follows the lines of the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let us remark that, although Theorems 12 and 13 are stated as purely existence result, the method exhibited in the proof of Theorem 12 is construc- tive, as shown in the following example, that we hope it helps to clarify the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Anyway, to compute the frame field using this method is not com- putationally efficient, since it requires to compute too many determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 11 Example 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In the case n = 3, N = 1, consider the matrix D(ε) = \uf8ee \uf8f0 ε 0 0 0 0 0 0 0 0 \uf8f9 \uf8fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' For U = R, this matrix satisfies rank(D(ε)) = � 1 if ε ̸= 0 0 if ε = 0 so we expect to obtain a frame field of solutions consisting in 2 vector, defined in R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The proof of Theorem 1 explains how to obtain one of the solutions, using certain cofactors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' This solution is x1(ε) = 1 ∥[0, ε, 0]T∥ \uf8ee \uf8f0 0 ε 0 \uf8f9 \uf8fb = \uf8ee \uf8f0 0 1 0 \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Now, following the idea in Theorem 13 we repeat the same proccess, but for the matrix B(ε) = D(ε) + x1(ε)x1(ε)T = \uf8ee \uf8f0 ε 0 0 0 1 0 0 0 0 \uf8f9 \uf8fb obtaining the solution x2(ε) = 1 ∥ ���� 0 0 1 0 ��� , ��� ε 0 0 0 ��� , ��� ε 0 0 1 ��� �T ∥ \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ���� 0 0 1 0 ���� ���� ε 0 0 0 ���� ���� ε 0 0 1 ���� \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = \uf8ee \uf8f0 0 0 1 \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Note that, in this case, the frame field can be extended to the whole open set U, although this may not happen in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Finally, note that this analytic frame field (x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk(ε)) obtained with this method, is not the only possible one satisfying the conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Remark 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Using Gram-Schmidt Method it is possible to find analytic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' in Cl(U)) vector fields �xk+1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , �xn(ε) in such a way that {x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk(ε), �xk+1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , �xn(ε)} (6) is an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let (y1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , yk(ε)) be a frame field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Then it satisfies the conditions if and only if, there exists a matrix K(ε) that maps the elements in the basis (x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk(ε)) into elements in the basis (y1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , yk(ε)) of the form K(ε) = P(ε) � T(ε) 0 0 In−k � P(ε)−1 (7) 12 where P(ε) is the matrix whose columns are the vectors in the base (6), T(ε) is a matrix which entries are analytic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' are in Cl(U)) and such that for each ε ∈ U belongs to the orthogonal group O(k) and finally In−k is the (n − k) × (n − k) identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Note that the entries of yi(ε) are analytic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' are in Cl(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' A matrix whose entries depend on some variables and belongs to O(n) for each value of these variables, such as K(ε), is sometimes called kinematic matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In fact, we can view the frame field (y1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , yk(ε)) as the result of applying a rigid motion to the original frame field (x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=', xk(ε)) in such a way that the entries in the matrix of this rigid motion have the adequate smoothness with respect to the variables in ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Linearly independent solutions for non-homogeneous linear systems From Theorem 9 and the results in the previous subsection, it is easy to prove the following: Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let ε0 ∈ R, and let U be a neighborhood of this ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let D(ε) = [γij(ε)]1≤i,j≤n, b(ε) = (b1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , bn(ε))T be such that, for 1 ≤ i, j ≤ n, γij(ε), bi(ε), are analytic for 1 ≤ i, j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Suppose that r = rank(D(ε0)) = rank([D(ε0) | b(ε0)]) and that for every ε ∈ U, rank([D(ε) | b(ε)]) ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let k = n − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Then there exist a neighborhood U∗ of ε0, an analytic vector field xp(ε) and an analytic frame field (x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk(ε)) such that any analytic vector field x(ε) satisfying Equation (1) can be writen as: x(ε) = xp(ε) + λ1(ε)x1(ε) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' + λk(ε)xk(ε) for some analytic functions λ1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , λn(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let ε0 ∈ RN, for N ≥ 1, and let U be a neighborhood of this ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let D(ε) = [γij(ε)]1≤i,j≤n, b(ε) = (b1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , bn(ε))T be such that, for 1 ≤ i, j ≤ n, γij(ε), bi(ε) are in Cl(U), for 1 ≤ i, j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Suppose that r = rank(D(ε0)) = rank([D(ε0) | b(ε0)]) and that for every ε ∈ U, rank([D(ε) | b(ε)]) ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let k = n − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Then there exist a neighborhood U∗ of ε0, a Cl(U∗) vector field xp(ε) and a Cl(U∗) frame field (x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk(ε)) such that any Cl(U∗) vector field x(ε) satisfying Equation (1) can be writen as: x(ε) = xp(ε) + λ1(ε)x1(ε) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' + λk(ε)xk(ε) for some functions λ1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , λn(ε) in Cl(U∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The existence of xp (the particular solution) is ensured by Theorem 9 and the existence of the frame fields (homogeneous solutions) by Theorems 12 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Sensitivity Analysis In this section we study Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' The first two subsections deal with the case in which rank(D(ε0)) = n − 1 and rank(D(ε)) ≤ n − 1 for ε in some neighborhood of ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In the first one, we present a direct method to solve Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In the second one, an adjoint method is provided to perform this same task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' This second type of methods are, computationally, more efficient for large values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Finally, in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='3 we discuss the difficulties for computation of sensitivities in the case rank(D(ε0)) < n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Direct Method Let us begin with the following straightforward observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Recall that we are considering solutions verifying x(ε0) = u, for a given unitary vector u, as in the formulation of Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let ε0 ∈ RN and U be a open neighborhood of ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let x(ε) be a C1(ε) vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Suppose that x(ε) is a solution of the system D(ε)x(ε) = b(ε) for ε ∈ U, where the entries in D(ε) and b(ε) are in C1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Then: D(ε0) ∂x ∂εi (ε0) = ∂b ∂εi (ε0) − ∂D ∂εi (ε0)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' (8) Proof: Equation (8) is obtained taking derivatives from D(ε)x(ε) = b(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' ✷ Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let ε0 ∈ RN and let U be a open neighborhood of ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let x(ε) be a C1(ε) vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' If ∥x(ε)∥ = 1 for ε ∈ U, then, for 1 ≤ i ≤ N: uT ∂x ∂εi (ε0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' (9) Proof: Equation (9) is obtained taking derivatives in x(ε)Tx(ε) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' ✷ Combining Equations (8) and (9) we can obtain a way to compute the derivatives inspired in Nelson’s method (that was originally developed for eigenvector sensitivity, see [4, 5, 22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Note that the general solution of Equa- tion (8) must be of the form ∂x ∂εi (ε0) = v + cu (10) for v being a particular solution of the system (8) and c a constant to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' From the fact that uT ∂x ∂εi (ε0) = 0 we obtain that: c = −vTu (11) So finally we get: 14 Direct Method for the homogeneous system D(ε)x(ε) = 0 such that x(ε0) = u and rank(D(ε)) = n − 1 is a neighborhood of ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='- We look for a particular solution v of the singular linear system D(ε0)v = − ∂D ∂εi (ε0)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='- Set c = −vTu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='- Finally, ∂x ∂εi (ε0) = v + cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Adjoint Method Suppose that, for some F : Rn → R, for some 1 ≤ j ≤ n, we want to compute ∂ ∂εj (F(x(ε0))) where x(ε) satisfies Equations (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Using this function F we obtain some notational advantages, we study (effortlessly) a more general problem and we can easily particularize this problem (taking F(x(ε)) = xi) to recover ∂xi ∂εj (ε0) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' For some p ∈ Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' λ ∈ R that do not depend on the variables in ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' consider the trivial equation: F(x(ε)) = F(x(ε)) + pTD(ε)x(ε) + 1 2λ(xT(ε)x(ε) − 1) � �� � =0 Taking derivatives we get: ∂ ∂εj (F(x(ε0))) = ∇F(x(ε0)) ∂x ∂εj (ε0) + pT � ∂D ∂εj (ε0)x(ε0) + D(ε0) ∂x ∂εj (ε0) � + λ � xT(ε) ∂x ∂εj (ε) � and re-organizing: ∂ ∂εj (F(x(ε0))) = pT ∂D ∂εj (ε0)x(ε0) + � ∇F(x(ε0)) + pTD(ε0) + λxT(ε) � � �� � (⋆⋆) ∂x ∂εj (ε0) To simplify the expression above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' we can choose pT and λ in such a way that (⋆⋆) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' To achieve this, we choose λ ∈ R to be the (unique) real number such that the following linear system has a solution (at this point is where we need that rank(D(ε)) = n − 1) and then we solve it to find p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' D(ε0)Tp = −λx(ε0) − ∇F(x(ε0))T (12) Putting these ideas together we obtain: Adjoint Method for the homogeneous system to compute partial derivatives of F(x(ε)) at ε0, where D(ε)x(ε) = 0 and x(ε0) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='- Find λ, p such that: D(ε0)Tp = −λu − ∇F(u)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='- ∂ ∂εj (F(x(ε0))) = pT ∂D ∂εj (ε0)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 15 Note that, p is not unique, that is, we can find two different solutions p1, p2 satisfying (12) and so (pT 1 − pT 2 )D = 0 (13) But the value of ∂ ∂εj (F(x(ε0))) does not vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' To check this, let us see that this invariance is equivalent to: (pT 1 − pT 2 )∂D ∂εj x = 0 and this equation is true as a consequence of Equations (8) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' If we want to compute several partial derivatives ∂x ∂εj1 (ε0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , ∂x ∂εjr (ε) the first part of the method is common to all of them and the second one just requires the computation of the corresponding derivative of D and a multipli- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Cases in which the solution of Problem 3 is not determined If for all ε ∈ U, rank(D(ε)) = n − k, for k > 1, then Problem 3 cannot be solved unless more information is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Let us see the following clarify- ing example: Example 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' For the case N = 1, n = 4, U = R and ε0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Consider the matrix D(ε) = \uf8ee \uf8ef\uf8ef\uf8f0 ε 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Consider some vector field x(ε) such that, for all ε ∈ R, D(ε)x(ε) = 0, x(0) = (0, 1, 0, 0)T This information is not sufficient to determine x′(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' To check this, just see that for every vector (0, 0, a, b)T in the linear space spanned by {(0, 0, 1, 0)T, (0, 0, 0, 1)T}, the solution x(ε) = (0, cos ε, sin aε, sin bε) satisfies x′(ε0) = (0, 0, a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In the case of eigenproblems this feature is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' It corresponds to the case in which λ(ε) is an eigenvalue of constant multiplicity h > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Suppose that we have guaranteed the existence of a frame field (x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk(ε)) consisting of k vector fields which are solutions of (2) and that some orthonor- mal vectors u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , uk are provided, so that it is prescribed that xj(ε0) = uj for j = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In this case, the derivative of the solutions must satisfy some extra conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' For 1 ≤ j1, j2 ≤ k, j1 ̸= j2: xj1(ε)xj2(ε) = 0 ⇒ uT j1 ∂x ∂εi (ε0) + uT j2 ∂x ∂εi (ε0) = 0 16 Even with this extra conditions, the solution to Problem (3) is not determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' It can be proved in a straightforward manner, studying the corresponding linear system, that has (k 2) degrees of freedom, or just noting the following: Remark 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' In the notation of Remark 15, see that for the frame field (x1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , xk(ε)) obtained following the method in the proof of Theorems 12 and 13 any other frame field (y1(ε), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' , yk(ε)) satisfies the corresponding conditions if and only if for 1 ≤ i ≤ j, yi(ε) = K(ε)xi(ε) (14) Now, if we impose that, for 1 ≤ i ≤ k, xi(ε0) = yi(ε0) = ui for some prescribed vector ui, then K(ε0) = In and so K(ε) is in SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Taking derivatives in Equation (14) we have that ∂yi ∂εi (ε0) = ∂K ∂εi (ε0)ui + ∂xi ∂εi (ε) If we want K(ε) to be of the type explained in Equation (7) and to satisfy K(ε0) = In we still have (k 2) free parameters since ∂T ∂εi (ε0) can be any skew-symmetric matrix of size k × k (see [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} +page_content=' Ackwnoledgements This work has been supported by the Spanish Agencia Estatal de Investi- gaci´on through project PID2020-116207GB-I00 and Junta de Comunidades de Castilla-La Mancha through project SBPLY/19/180501/000110.' metadata={'source': 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19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfwzUG/content/2301.13164v1.pdf'} diff --git a/c9E0T4oBgHgl3EQf5AL3/content/tmp_files/2301.02747v1.pdf.txt b/c9E0T4oBgHgl3EQf5AL3/content/tmp_files/2301.02747v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a8dfea654707de1733a189dce4f77ba60ef126ca --- /dev/null +++ b/c9E0T4oBgHgl3EQf5AL3/content/tmp_files/2301.02747v1.pdf.txt @@ -0,0 +1,1345 @@ +Modeling Scattering Coefficients in Antenna Design using Self-Attentive +Complex Polynomials with Image-based Representation +Andrew Cohen∗ 1 Weiping Dou 1 Jiang Zhu 1 Slawomir Koziel 2 Peter Renner 1 Jan-Ove Mattsson 1 +Xiaomeng Yang 1 Beidi Chen 1 Kevin Stone 1 Yuandong Tian∗ 1 +Abstract +Finding antenna designs that satisfy frequency +requirements and are also optimal with respect +to multiple physical criteria is a critical compo- +nent in designing next generation hardware. How- +ever, such a process is non-trivial because the +objective function is typically highly nonlinear +and sensitive to subtle design change. Moreover, +the objective to be optimized often involves elec- +tromagnetic (EM) simulations, which is slow and +expensive with commercial simulation software. +In this work, we propose a sample-efficient and +accurate surrogate model, named CZP (Constant +Zeros Poles), to directly estimate the scattering +coefficients in the frequency domain of a given +2D planar antenna design, without using a simu- +lator. CZP achieves this by predicting the com- +plex zeros and poles for the frequency response +of scattering coefficients, which we have theo- +retically justified for any linear PDE, including +Maxwell’s equations. Moreover, instead of using +low-dimensional representations, CZP leverages +a novel image-based representation for antenna +topology inspired by the existing mesh-based EM +simulation techniques, and attention-based neural +network architectures. We demonstrate experi- +mentally that CZP not only outperforms baselines +in terms of test loss, but also is able to find 2D an- +tenna designs verifiable by commercial software +with only 40k training samples, when coupling +with advanced sequential search techniques like +reinforcement learning. +1. Introduction +The next generation of Metaverse computing devices such +as virtual reality (VR) and augmented reality (AR) offers +*Equal contribution +1Meta AI 2Reykjavic University. Corre- +spondence to: Andrew Cohen , Yuan- +dong Tian . +Preprint. Work in Progress +exciting possibilities for immersion in the digital world. In +pursuit of seamless wireless connectivity with high wire- +less throughput and low latency to enable such user experi- +ences, antenna design has become even more challenging +than ever before. This is because the antenna’s physical +volume has been constrained by the stylish, light-weight +industrial design requirements for wearable devices, while +the demand for broader spectrum coverage to support the +AR/VR experiences has been ever growing. To meet the +demand, one device contains up to 10 to 20 antennas in a +small form factor. Moreover, the recent adoption of WiFi 6E +in the consumer electronic products such as Meta’s Quest +2, extends the WiFi high band to the entire 6GHz spectrum, +which presents a significant new challenge in antenna design. +From the wireless industry perspective, there is a demand +to design and optimize antennas that can unleash their full +potential under the given space constraints. +Computational antenna design heavily relies on full-wave +electromagnetic (EM) simulations. The process is expensive +in terms of both simulation and engineer time as antenna +engineers often iterate on antenna configurations using CPU- +intensive commercial software (CST, 2021; XFD). This high +computational overhead is the primary bottleneck in an an- +tenna engineer’s ability to rapidly experiment with different +antenna geometric structures. A single simulation can take +dozens of seconds to several weeks depending on the sys- +tematic complexity of a device. And this learning process +is sequential because insights gained from simulating one +set of candidate topologies are then used for generating the +next set of candidates. +Commercial antenna software often uses mesh-based EM +simulation. It involves finding a suitable mesh data struc- +ture to convert the underlying Maxwell partial differential +equations (PDEs) of a system into ordinary differential equa- +tions (ODEs) that can be solved by finite difference methods +(FDMs) (Yee, 1966). The mesh is often non-uniform, and +typically with much higher resolution in regions of impor- +tance such as the boundaries of and borders between ma- +terials. It is not uncommon for a mesh to contain tens of +millions of cells, leading to expensive and slow computa- +tion. +arXiv:2301.02747v1 [cs.LG] 6 Jan 2023 + +Modeling Scattering Coefficients in Antenna Design using Self-Attentive Complex Polynomials with Image-based +Representation +Developing cheaper alternatives to simulation is the focus +of extensive research. Many works attempt to replace costly +EM simulation with a lighter weight surrogate model, which +trades off some degree of accuracy for computational effi- +ciency. The surrogate model is then used by a downstream +optimization procedure. In the EM based microwave circuit +design, such as microwave filters, impedance-matching net- +works, multiplexers, etc., the equivalent-circuit, empirical, +and semi-analytical models and combinations have been +used as surrogate models with the links to the full-wave sim- +ulation (Rayas-Sanchez, 2004; Bakr et al., 2000). The same +approaches have been rarely applied to antenna modeling, +due to the fact that the radiating structures are too com- +plex to lend themselves to analytical and/or circuit model- +ing. Broadly, there are two approaches to antenna surrogate +modeling, coarser approximate physics-driven simulation +(Zhu et al., 2007; Koziel & Ogurtsov, 2013) or data-driven +methods which model the computation performed by the +simulator (Koziel, 2017). In this work, we focus on the +latter and leverage recent advances in machine learning to +efficiently learn a surrogate model. It has been noted that +machine learning has great promise which has not yet been +fully realized in antenna design (Dou et al., 2022) . +There are many challenges to learning a useful surrogate +model. Theoretically, the antenna search space is high +dimensional and the relationship between antenna topol- +ogy and resonance profile is highly non-linear. However, +state-of-the-art neural network architectures have demon- +strated the ability to learn complicated relationships in +many domains such as language (Brown et al., 2020), +images (Ramesh et al., 2021) and molecules and pro- +teins (Chilingaryan et al., 2022). Additionally, in the an- +tenna space the cost of data is significant since it is typically +collected from the commercial EM software which is com- +putationally expensive. +In this work, we propose a novel surrogate model network ar- +chitecture which addresses both difficulties raised above. It +leverages a transformer-based encoder (Vaswani et al., 2017) +to deal with the highly non-linear relationship of antenna +topology and resonances and also incorporates domain- +specific inductive biases to alleviate the issues posed by +limited data. Here, we focus on the scattering coefficients of +antennas, specifically the reflection coefficient which is used +to describe an antenna’s resonant frequencies and operating +bandwidth. The contributions of this work are 4-fold: +• Inspired by state-of-the-art mesh-based simulation +techniques, we propose a novel image-based repre- +sentation of antennas geometries that is both sample- +efficient and captures the critical boundary information +that is captured by high resolution meshes. +• We show that the resonance characteristic of an antenna +can be written analytically as the ratio of two complex- +valued polynomials with a compact representation in +terms of a constant, zeros and poles. Furthermore, this +property holds for any linear PDE, beyond Maxwell’s +equations. +• We propose a transformer architecture that first tok- +enizes the image representation and then directly pre- +dicts the complex valued zeros and poles of the S +scattering matrix to compute the frequency response +which we refer to as CZP. +• Experiments demonstrate the superiority of the CZP +model over multiple baselines on test set loss. Fur- +thermore, when coupled with an optimization proce- +dure like reinforcement learning, CZP can find antenna +topologies that are verified by commercial EM model- +ing software to meet design specifications. This shows +that CZP not only generalizes to unseen designs, but +also is robust to potentially adversarial designs found +by the optimization procedure. +2. Antenna Preliminaries +In this section, we define key notions and concepts and also +introduce the antenna design problem. Then, we present +a derivation based on how traditional EM solvers compute +the solution to Maxwell’s equations and is the theoretical +innovation on which we base our CZP (Constant Zeros +Poles) architecture. +2.1. Antenna Parameterization +In this work, we consider 2D planar antennas to demonstrate +the approach. An antenna design consists of the following +components: +Substrate. The rectangular printed circuitry board of width +Sx and height Sy on which the other components sit. The +substrate has thickness Sz and dielectric permittivity ϵr. +Ground plane. A solid rectangle extending through the +entire substrate in the x direction and partially in the y +direction. +Discrete port. The port location is the coordinate px, py +and is dependent on one of the front-side metallic patches. +Front-side metallic patches. The antenna contains M rect- +angular metallic patches which can freely move within the +substrate area or pre-determined ranges. The m-th patch +pm is defined by its width and height sm,x, sm,y and, the +coordinate of the bottom left corner lm,x, lm,y. When the +boundary of a metallic patch goes beyond the substrate, the +excess is simply clipped. When patches overlap, there is +no increase in the thickness; they combine to make a single +metallic patch that is no longer rectangular. +Combining all these specifications, we now have an overall + +Modeling Scattering Coefficients in Antenna Design using Self-Attentive Complex Polynomials with Image-based +Representation +Figure 1. Top: An instance of an antenna from the five patch ex- +ample in a AR device. Yellow corresponds to patches of metallic +substrate and purple corresponds to the board on which the antenna +sits. Bottom: The corresponding frequency response of the given +antenna. +design choice vector defined as +h = {Sx, Sy, Sz, px, py, {sm,x, sm,y, lm,x, lm,y}M +m=1}, +(1) +which determines the antenna’s frequency response de- +scribed by the logarithm of modulus of the scattering coef- +ficients log |S11(ω)|, typically expressed in decibels (dB). +S11(ω), as a function of frequency ω, is defined as the fol- +lowing (Caspers, 2012): +S11(ω) := Zin(ω)/Z0 − 1 +Zin(ω)/Z0 + 1 +(2) +where Zin(ω) is the input impedance of the antenna, deter- +mined by the design vector h. Figure 1 shows an example +antenna design and its S11(ω). +The objective. Antenna engineers aim to find the right +design choice h so that specific antenna design targets are +met over the frequency bands of interest, for example, there +needs to be dips (i.e., more absorption) in S11(ω) at WiFi +2.4GHz band and WiFi 5-7GHz band for WiFi 6E, as shown +in Figure 1 bottom row. +Five Patch Example. In this work, we consider an antenna +example with an FR-4 substrate that is 30mm by 6mm and 5 +front-side metallic patches with fixed dimensions and loca- +tion boundaries (see Appendix for details). Additionally, we +assume that the only degrees of freedom are the locations +{sm,x, sm,y} of each of the 5 patches as defined by the coor- +dinates of the bottom left corner. We constrain the problem +as such for experimental simplicity and acknowledge this is +a simplified setting with respect to production-tier antenna +optimization. However, the proposed surrogate model is +agnostic to the assumption on patches and the optimisation +procedure can be easily extended to variable patches of +varying dimensions. +3. An analytical formula of scattering +coefficients S11(ω) +The relationship between an antenna’s topology and its re- +sponses is governed by the well-known Maxwell’s equations +which can be written in the form of PDEs. When solv- +ing Maxwell’s equations, traditional methods discretize the +space and convert the PDEs into the following ODEs (Wei- +land, 1977): +˙φ = A(h)φ +(3) +where the vector φ contain electromagnetic quantities (e.g., +electric/magnitude field strength and potentials, voltages +and currents, etc) at each grid cell. Note that φ is indexed +by spatial location x: φ = [φ(x1), φ(x2), . . .] and thus is +an infinite-dimensional vector in the continuous case and a +vector of dimension N in finite difference methods. Finally, +A(h) is a matrix of size N-by-N related to material proper- +ties h determined by the design, and topological structure +of the discretized grid cells. +For better understanding, here we put a concrete ex- +ample of Eqn. 3. +Consider a one-dimensional wave +equation +∂2ψ +∂t2 += +c2 ∂2ψ +∂x2 . +Then by setting φ += +[ψ(x1), . . . , ψ(xN), ∂ψ +∂t (x1), . . . , ∂ψ +∂t (xN)]⊤ ∈ R2N, the +wave equation can be written in the form of Eqn. 3 with +A = +� +0 +1 +c2B +0 +� +, +where B ∈ RN×N spatially discretizes the operator +∂2 +∂x2 . +From the initial condition φ(x, 0), classic techniques (e.g., +finite element methods (Weiland, 1977)) simply perform +temporal integration to get the spatial-temporal signal +φ(x, t), and compute S11(ω) by its definition, which is +time-consuming. +Instead, we choose to follow another path, by directly per- +forming a temporal Fourier transform for ODEs. Following +this, we show that S11(ω) has an analytic form as the quo- +tient of two complex polynomials with respect to frequency +ω of the same order. This is presented formally in the fol- +lowing theorem: +Theorem 3.1 (Analytical Structure of Scattering Coeffi- +cients). For ODE in the form of Eqn. 3, if A(h) is diago- +nalizable, then the logarithm of modulus of the scattering +coefficients log |S11(ω)| can be written as: +log |S11(ω)| = log |c0(h)| + +K +� +k=1 +log |ω − zk(h)| +|ω − pk(h)| +(4) +where the constant c0(h), zeros {zk(h)}K +k=1 and poles + +10 +a +-15 +S +-20 +-25 +2 +0 +3 +4 +5 +6 +1 +Freguency (GHz)Modeling Scattering Coefficients in Antenna Design using Self-Attentive Complex Polynomials with Image-based +Representation +{pk(h)}K +k=1 are complex numbers and all depend on A(h) +and thus functions of the design choice h. +This motivates us to train a neural network to predict the +zeros zk (i.e., roots of the nominator) and poles pk (i.e., roots +of the denominator), as well as a global constant c0 from +material properties h, in order to predict scatter coefficients +S11 that are provided by existing commercial software as a +supervision. +With this formulation, we avoid any forward numerical inte- +gration and arrive at the quantity we want in one inference +pass. Again, we point out that our approach is general and +not restricted to antenna design; For any linear PDE sys- +tem that can be discretized into the form of Eqn. 3 where +A(h) does not vary with φ, its frequency response can be +computed similarly. +4. Network Architecture +In this section, we discuss the details of the proposed model. +Specifically, we propose a novel image representation for a +2D antenna inspired by the mesh representations commonly +used by EM simulators. Then, following the analysis in the +previous section, we introduce our image-based transformer +architecture which predicts the zeros and poles of scattering +coefficients. +4.1. Image representation +Mesh-based finite element methods underpin many of the +available simulation tools in electromagnetics and other +fields (Pardo et al., 2007). The mesh converts the underlying +PDEs of the system into an ODE solvable by finite element +methods (Weiland, 1977). Mesh representations use the +fact that an antenna’s resonance characteristics are directly +related to its local and global topological structure. This +motivates the use of images for learning a surrogate model +as it contains the same local and global spatial information. +A model would have to cope with a naive representation (i.e. +the coordinates of front-side metallic patches) by learning +these spatial relationships. +A critical component of successful meshing is to generate +non-uniform, adaptive meshes which allocate high resolu- +tion, dense meshing to areas in which the quantity φ may +change rapidly (e.g., at sharp corners). Adaptive meshing +enables the simulation of systems unsolvable by traditional +discretization methods (Pfaff et al., 2021). Guided by this, +we posit that an image representation of an antenna should +provide the key regions (i.e., boundaries and corners of sub- +strate) explicitly so that a neural network does not need to +spend unnecessary computation learning these features. +We propose a three channel image representation. The first +two channels provide the boundary locations in the x and +y directions where pixel values v ∈ [0, 1] are floating point +to represent the distance to the nearest pixel in the x or +y directions, respectively. For example, given the bottom +left (xbl, ybl) and top right (xtr, ytr) floating-number coor- +dinates of a rectangular patch, we compute the pixel indices +as the floor, +¯xbl = ⌊xbl⌋, ¯ybl = ⌊ybl⌋, ¯xtr = ⌊xtr⌋, ¯ytr = ⌊ytr⌋. +Then, we compute the values +vl = 1 − (xbl − ¯xbl), vr = xtr − ¯xtr +vb = 1 − (ybl − ¯ybl), vt = ytr − ¯ytr +where vl, vr,vb,vt correspond to the left, right, bottom and +top boundary values, respectively. The left/bottom boundary +is subtracted from 1 where as the right/top is not because +the floor function has a subtly different semantic meaning +between these cases; Without the loss of generality, the +floor of the left/bottom boundary is not contained inside the +interior of the patch whereas the floor of the right/top is. We +chose this design as it enables sensible image dimensions +(i.e. 60 × 300 for a 6mm ×30mm image with a resolution +of 10 pixels to 1mm) however others are possible. Finally, +note, separating the x and y boundaries into two channels +enables explicit representation of the corners of patches. +Finally, a third channel provides the interior of the antenna +as a binary image where v = 1 for all index pairs x, y such +that x ∈ [¯xbl + 1, ¯xtr − 1], y ∈ [¯ybl + 1, ¯ytr − 1] and v = 1. +Please see Algorithm 1 for pseudocode of the process and +Figure 2 for an example image. Some details are omitted +such as patch dimensions which go beyond the board or +overlapping patches but these are straightforwardly handled +via clipping and masking. +Algorithm 1 ADD PATCH TO IMAGE(xbl, ybl, xtr, ytr, +image) +Input: xbl, ybl, xtr, ytr: floating point bottom-left and +top-right coordinates of rectangular patch, image: Image +array +1: // Lower bound on coordinates to index image +2: ¯xbl = ⌊xbl⌋, ¯ybl = ⌊ybl⌋, ¯xtr = ⌊xtr⌋, ¯ytr = ⌊ytr⌋ +3: // Write X boundaries +4: imagex bound[¯ybl : ¯ytr][¯xbl] = 1 − (xbl − ¯xbl) +5: imagex bound[¯ybl : ¯ytr][¯xtr] = xtr − ¯xtr +6: // Write Y boundaries +7: imagey bound[¯ybl][¯xbl : ¯xtr] = 1 − (ybl − ¯ybl) +8: imagey bound[¯ytr][¯xtr : ¯xtr] = ytr − ¯ytr +9: // Write interior +10: imageinterior[¯ybl+1 : ¯ytr−1][¯xtr+1 : ¯xtr−1] = 1.0 + +Modeling Scattering Coefficients in Antenna Design using Self-Attentive Complex Polynomials with Image-based +Representation +Figure 2. Three channel image representation. Top: Boundary +values represent distance to nearest pixel in the x-direction. Mid- +dle: Boundary values represent distance to nearest pixel in the +y-direction. Bottom: Binary interior of the antenna. This channel +does not contain boundaries. +4.2. Surrogate Model +In this section, we propose an architecture for a surrogate +model which predicts the zeros and poles directly from +the image representation which is then used to compute +scattering coefficients. The architecture is based on the +Visual Transformer (Wu et al., 2020) which is motivated by +the insight that local, spatial components such as boundaries +between substrate of the antenna should be tokenized and +then used by a transformer (Vaswani et al., 2017) to compute +the global characteristics. +Given the input image I = I(h) ∈ R3×HW as a function +of design choice h, we first augment it with two additional +channels of linearly spaced x and y coordinates (Liu et al., +2018), to yield augmented image ˆI ∈ R5×HW . This is +because the specific location of antenna components, in +addition to its topology, determines the corresponding fre- +quency response. Then, a CNN takes ˆI as an input and +generates feature maps X ∈ RHW ×C where H, W and C +are the height, width and channel dimension, respectively. +A filter-based tokenizer (Zhang et al., 2019) generates L +visual tokens T ∈ RL×C by mapping each pixel via a point- +wise convolution to L groups with matrix W ∈ RC×L and +computes a softmax in the pixel dimension +A = SoftmaxHW (XW) +where A ∈ RHW ×L is referred to as an attention map. Vi- +sual tokens T are computed via T = AT X which is the +weighted average of pixels in the original feature map X. +Intuitively, the tokens T capture semantics such as relative +boundary and corner locations and from this the transformer +computes the global characteristic of the antenna configu- +ration. Please see Figure 9 in Appendix D for a subset of +the learned attention maps for a specific antenna instance +which demonstrate this. +After that, T is then passed through a multi-layer trans- +former encoder (Vaswani et al., 2017), flattened and passed +through a fully connected layer and a non-linearity. From +this representation, three separate complex-valued fully con- +nected layers predict the constant, zeros and poles. Con- +cretely, let Cθ, Zθ, Pθ be linear layers parameterized by θ. +Then, +v = FC(Transformer(T)) +c0(h) := Cθ(v), z(h) := Zθ(v), p(h) := Pθ(v) +where h is the design choice and c0(h), z(h), p(h) are +the constant and vectors (of length K) of zeros and poles, +respectively, used to compute the frequency response as per +Equation 4. We refer to this architecture which outputs the +constant, zeros and poles as CZP. +4.3. Model training +When training CZP models, we do not have direct super- +vision to c0(h), zeros z(h), and poles p(h), but only the +S11(ω) provided with CST Microwave Studio (CST, 2021) +as ground truth. Therefore, we leverage Eqn. 4 to compute +estimated S11(ω) with c0, z, and p, so that it can match the +ground truth. We then train the model via back-propagation +in an end-to-end manner to minimize the Mean Squared +Error (MSE), for frequencies in the range [0.2 − 7.0]GHz at +increments of 0.1 (i.e., 69 dimensions). We use a shrinkage +loss (Li et al., 2018) variant of MSE as we found that with +vanilla MSE, the model had higher error on the crucial parts +of the scattering coefficients (i.e., the resonances). +5. Experiments +In this section, we demonstrate the impact of our architec- +tural choices and image representation on the five patch +example antenna discussed in Section 2.1. Specifically, we +show that: +• Our proposed image representation is a significant im- +provement over reasonable coordinate-based inputs as +well as a naive binary image input. +• The CZP formulation outperforms raw prediction +when using the same transformer architecture proposed +in Section 4 and the proposed image representation. +• The transformer architecture outperforms a CNN for +the image representation and an MLP for coordinate +based input. +• CZP generalizes well to unseen antenna designs, not +only on a held-out dataset, but also as a surrogate model + +Modeling Scattering Coefficients in Antenna Design using Self-Attentive Complex Polynomials with Image-based +Representation +for designs proposed by the reinforcement learning +(RL) based search procedure, as verified to meet spe- +cific resonance requirements by commercial softwares. +5.1. Surrogate Modeling +We use 48K total samples, uniformly sampled and simu- +lated with CST Microwave Studio (CST, 2021) where each +sample takes between 90 and 120 seconds to simulate. 90% +of samples are used for training and 10% are used for testing. +From the training set, 10% are randomly sampled and used +for validation. Each experiment is run for 3 random seeds. +Appendix C provides all experimental hyperparameters. +Figure 3 illustrates the first set of experiments in which +we demonstrate the effectiveness of (1) our novel image +representation and (2) the proposed CZP when using the +transformer architecture. To show (1), we compare against +a coordinate-based method which concatenates the normal- +ized bottom-left x, y coordinate of each patch with a one- +hot vector to distinguish between patches. When using the +transformer architecture, this generates 5 tokens, each with +dimension 7. Before being processed by the transformer, +each token is projected into a 256 dimensional vector by a +2-layer MLP with hidden layer of width 256. Additionally, +to demonstrate (2), for both coordinate and image input, we +compare against directly predicting the raw 69 dimensional +frequency response with a fully connected layer, referred +to as Raw in figures. Additionally, we ablate over different +degree K of CZP with values 8, 12, 16, 20 and the number +of attention layers L with values 8, 6, 4, 2. +First, within each configuration, the image representation +improves over its coordinate counterpart by a minimum of +9.6% with L = 4 and raw prediction and a maximum of +26.8% with L = 2 and K = 12. Second, for the image repre- +sentation, CZP improves over raw prediction by a minimum +of 9.2% with L = 8 and K = 12 and a maximum of 28.0% +with L = 2 and K = 12. For the coordinate representation, +CZP improves over raw prediction by a minimum of 4.6% +with L = 8 and K = 12 and a maximum of 25.1% with +L = 2 and K = 8. Finally, increasing the transformer depth +from 2 to 8 layers improves raw prediction for image and +coordinate representations by 29.5% and 35.9%, respec- +tively. Increased depth improves the CZP an average of +15.4% and 20.9% for image and coordinate representations, +respectively. +From these statistics, we can extract the following insights +which support the CZP architecture and image represen- +tation as powerful inductive biases; (1) With fewer trans- +former layers, CZP yields greater improvement over raw +prediction and (2) CZP and image representation benefit +the least from increasing the complexity of the model and, +at the opposite extreme, raw prediction and coordinate rep- +resentation benefit the most. These two points show that +without these inductive biases, deeper models are required +as shallower models are likely to fall into local minima. +In Figure 4, we provide results for 4 other baselines to +show the impact of the transformer and image representation +over reasonable alternatives e.g., a fully connected MLP +with coordinate input, a CNN with our image input, the +transformer with a naive single-channel binary image input, +and the Fourier Neural Operator (FNO) (Li et al., 2020) +developed to solve other PDEs such as Navier-Stokes with +image input. In the last row we reproduce the results of +the 8-layer transformer with image input from Figure 3. +The 8 layer transformer is a 40%+ improvement on these +baselines. +Finally, in Figure 5, we provide a data ablation with raw +prediction and CZP K = 20. The trend of CZP out per- +forming raw prediction holds in this setting as well although +the differences in test loss are small. However, in the next +section, we show that our model greatly outperforms the +baselines when used for optimization when trained with +less data, more robust for unseen designs. For other qualita- +tive results such as attention map visualizations, please see +Appendix D. +5.2. Optimization +In this section, we demonstrate the utility of the proposed +model by showing it can be used by an optimization proce- +dure to find antenna configurations that have specific reso- +nance characteristics. This is a significant test of the gener- +alization and robustness of the model since (1) an antenna +with the desired resonances is not contained in the training +set and (2) an optimization procedure can very easily find +adversarial configurations to exploit the weaknesses of the +surrogate model (Yuan et al., 2017). We hypothesize that +CZP will be far more robust than raw prediction to these +kinds of samples because it is by design smooth (i.e., a +ratio of two polynomials) whereas raw prediction has no +built in bias encouraging this property. Please see Figure 8 +in Appendix D for qualitative intuition regarding this. In +this section, we provide results which demonstrate that our +proposed model, when used by an optimization procedure, +has a significantly higher success rate and is more robust to +dataset size than the baseline. +We frame antenna design as a reinforcement learning +(RL) (Sutton & Barto, 1998) problem where an agent is +tasked with sequentially placing each of the 5 patches +such that the frequency response of the final antenna meets +the resonance characteristics. Recall from Section 2, this +means that the corresponding S11 is below a certain thresh- +old at specific frequency ranges. In this problem, the fre- +quency ranges are 2.4 GHz-2.5 GHz and 5.1 GHz-7.0GHz +and the target thresholds are t[2.4−2.5] = −6.0 dB and +t[5.1−7.0] = −6.0 dB, the spectrum for WiFi 6E. + +Modeling Scattering Coefficients in Antenna Design using Self-Attentive Complex Polynomials with Image-based +Representation +Layers +Input Type +Raw +CZP K = 8 +CZP K = 12 +CZP K = 16 +CZP K = 20 +L = 8 +Image +.00284 ± 7e−5 +.00243 ± 1e−4 +.00258 ± 3e−5 +.00253 ± 5e−5 +.00234 ± 2e−5 +Coord +.00327 ± 5e−5 +.003 ± 8e−5 +.00312 ± 4e−5 +.00307 ± 4e−5 +.00303 ± 5e−5 +L = 6 +Image +.00312 ± 9e−5 +.00254 ± 3e−5 +.00253 ± 9e−5 +.00255 ± 6e−5 +.00249 ± 8e−5 +Coord +.00357 ± 8e−5 +.00308 ± 9e−5 +.00309 ± 7e−5 +.00313 ± 8e−5 +.00303 ± 7e−5 +L = 4 +Image +.00348 ± 1e−4 +.00268 ± 7e−5 +.00266 ± 2e−4 +.00251 ± 5e−5 +.00252 ± 1e−4 +Coord +.00385 ± 5e−5 +.00317 ± 6e−5 +.00328 ± 8e−5 +.0032 ± 1e−4 +.00322 ± 2e−5 +L = 2 +Image +.00403 ± 4e−5 +.00292 ± 1e−4 +.0029 ± 1e−4 +.00294 ± 1e−4 +.00292 ± 2e−4 +Coord +.0051 ± 8e−5 +.00382 ± 2e−4 +.00396 ± 1e−4 +.00385 ± 5e−5 +.00384 ± 9e−5 +Figure 3. Mean and standard deviation of the test loss over 3 seeds with the transformer architecture for image and coordinate input +representations and L = 8, 6, 4, 2 attention layers. Results are reported for raw frequency prediction and the CZP architecture with degree +K = 8, 12, 16, 20. In all configurations, the image representation outperforms coordinates and CZP outperforms raw prediction. +Arch + Input +Raw +CZP K = 8 +CZP K = 12 +CZP K = 16 +CZP K = 20 +MLP + Coord +.00492 ± 5e−5 +.00502 ± 1e−4 +.00553 ± 3e−4 +0.00507 ± 3e−4 +failed +CNN + Image +.0054 ± 3e−5 +.00496 ± 1e−3 +.00405 ± 1e−4 +.00424 ± 1e−4 +failed +Transformer + +.0049 ± 1e−4 +.005 ± 2e−4 +.00488 ± 9e−5 +.00501 ± 1e−4 +.0049 ± 8e−5 +Binary Image +FNO + +.00724 ± 5e−5 +.00715 ± 1e−4 +.00706 ± 5e−5 +0.0073 ± 2−4 +.00724 ± 9−5 +Image +Transformer + +.00284 ± 7e−5 +.00243 ± 1e−4 +.00258 ± 3e−5 +.00253 ± 5e−5 +.00234 ± 2e−5 +Image +Figure 4. Mean and standard deviation of the test loss over 3 seeds for ablations of architectural components of the proposed model and +baselines. Results reported are for raw prediction and CZP with degree K = 8, 12, 16, 20 for the following configurations: 6-layer MLP +with coordinate input, 5 layer CNN with image input, and 8 layer transformer with binary image input and 4-layer FNO with image input. +% Training +Transformer Out +Data +Raw +CZP K = 20 +25% +.00621 ± 3e−4 +.00574 ± 1e−4 +50% +.00387 ± 2e−4 +.0038 ± 2e−4 +75% +.00329 ± 1e−4 +.00296 ± 2e−4 +100% +.00284 ± 7e−5 +.00234 ± 2e−5 +Figure 5. Mean and standard deviation of the test loss over 3 seeds +for CZP K = 20 and raw prediction with the 8-layer transformer +architecture and image input with randomly sampled subsets of the +training data for portions 25%, 50% and 75%. CZP has a lower +test loss. +Formally, we define the state and action of the Markov +Decision Process (MDP) (Puterman, 1994) as: +• State: A one-hot identifier and (x, y) coordinates of +the bottom left corner of the patches which have been +placed and a one-hot vector for the next patch to be +placed. +• Action: (x, y) coordinates of the bottom-left corner of +the next patch to be placed. +After all patches have been placed, the coordinates are con- +verted to the image representation and the surrogate model +predicts the frequency response log |S11(ω)|. From the final +log |S11(ω)|, we compute the following reward components +for each resonance target. +r[2.4−2.5] = min(t[2.4−2.5] − log |S11(ω)|[2.4−2.5]) +(5) +r[5.1−7.0] = min(t[5.1−7.0] − log |S11(ω)|[5.1−7.0]) +(6) +where the subscripts correspond to list slicing. The sum +r = r[2.4−2.5]+min(1.0, r[5.1−7.0]) is then the reward given +at the final timestep and at all previous timesteps the reward +is zero. Note, we prevent the second reward component +from being greater than 1.0 because in experiments the +higher band (5.1 GHz-7.0 GHz) seemed to be easier to +optimize and often led to local minima that did not optimize +the lower band (2.4 GHz-2.5 GHz). To optimize, we use +the implementation of Soft Actor Critic (SAC) (Haarnoja +et al., 2018) from Stable-Baselines3 (Raffin et al., 2021) and +build the environment using the Gym API (Brockman et al., +2016). Default hyperparameters are used except we perform +two updates at the end of each episode as opposed to one or +more updates per step. +For these experiments, we use the CZP K = 20 and raw +prediction architectures with an 8-layer transformer as these +achieved the lowest test losses. For each of the 3 seeds for +each architecture trained in the previous section, we run 3 +seeds of RL optimization for a total of 9 experiments per +configuration. In each experiment, we deploy the SAC agent + +Modeling Scattering Coefficients in Antenna Design using Self-Attentive Complex Polynomials with Image-based +Representation +(a) CZP +(b) Raw +Figure 6. Two successful antenna configurations (top row) and corresponding frequency responses (bottom row) predicted by the model +(red) and computed by CST (green) found by optimization via RL for (a) CZP and (b) Raw. +% Data +Transformer Out +Raw +CZP K = 20 +% Top 3 +Any Top 3 +% Top 3 +Any Top 3 +25% +11.1% +22.2% +33.3% +55.6% +50% +14.8% +22.2% +40.7% +66.7% +75% +33.3% +55.6% +55.6% +77.8% +100% +51.9% +66.7% +88.9% +100% +Figure 7. Success rate for the % of top 3 configurations and if +any of the top 3 configurations meet design requirements. Exper- +iments performed for 3 seeds for RL for each of the 3 seeds of +CZP K = 20 and raw prediction with the 8-layer transformer ar- +chitecture and image input from the previous section. Experiments +conducted also with randomly sampled subsets of the training data +for portions 25%, 50% and 75%. CZP is more robust than raw +prediction to the optimization procedure and to dataset size. +for 25K total episodes or 125K total timesteps (since the +agent places 1 of 5 patches each step). We also investigate +the robustness of this process to dataset size which is critical +in the domain of antenna design as sample collection is +expensive. +Generally, SAC is able to find antenna configurations which +meet the requirements when using CZP and raw predic- +tion architectures and in Figure 6 we provide examples in +columns (a) and (b), respectively. The top row provides +the found antenna configuration and the bottom row the +frequency responses predicted by the model (red) and the +CST simulation (green). +However, in terms of success rate (i.e., how many config- +urations or optimization runs actually produce an antenna +which meets the constraints), CZP significantly out per- +forms raw prediction. Specifically, in Figure 7, we provide +the percentage of the top 3 configurations (i.e., 3 per seed for +a total of 27) found over all seeds which meet the constraints +and also the percentage of runs where any of the top 3 meet +the constraints. Additionally, we perform a data ablation to +show that CZP is more robust to less data demonstrating its +strength as an inductive bias. +6. Conclusion +In this work, we presented a novel surrogate model architec- +ture to be used by optimization techniques for the problem +antenna design. We first derived a theoretical form for +the S11 scatter coefficients based on which we proposed +the CZP network architecture. Additionally, as input to +the proposed surrogate model, we proposed a novel image +representation inspired by existing mesh-based simulation +techniques. We then demonstrated experimentally that the +proposed model and image representation are significant +advances through architecture and data ablation studies. Fi- +nally, we showed that the proposed surrogate model had +significantly higher utility in terms of success rate for opti- +mization of antenna design than baselines. +Although the results are significant, the problem investigated +in this work is still relatively simple compared to production +level antenna systems. Future work will involve solving +more complicated 2D problems as well as generalizing the +proposed model and image representation to 3D antenna. +Additionally, in this line of work, we plan to explore other +tokenization schemes that are as information rich as images +but are more computationally efficient since images require +convolutions to featurize. Lastly, future work will involve +the application to linear PDEs in general. + +Prediction +￥￥￥**** +-0 +Simulation +Target +-5 +-10- +B +p) +-15 +S +-20 - +-25 - +-30 - +6 +7 +0 +L +2 +3 +4 +5 +Frequency (GHz)Prediction +0- +Simulation +Target +-2 - +-4 - +B +p) +-8 +-10 - +-12 +-14 - +-16- +0 +1 +2 +3 +4 +5 +6 +7 +Frequency (GHz)Modeling Scattering Coefficients in Antenna Design using Self-Attentive Complex Polynomials with Image-based +Representation +References +Remcom. +URL +https://www.remcom.com/ +xfdtd-3d-em-simulation-software/. +CST Studio Suite, +2021. +URL +https://www. +3ds.com/products-services/simulia/ +products/. +Bakr, M. H., Bandler, J. W., Madsen, K., and Rayas- +Sanchez, J. E. Space-mapping optimization of microwave +circuits exploiting surrogate models. IEEE Transactions +on Microwave Theory and Techniques, 48:2297–2306, +2000. +Brockman, G., Cheung, V., Pettersson, L., Schneider, J., +Schulman, J., Tang, J., and Zaremba, W. Openai gym. +2016. +Brown, T. 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K., and Koziel, S. +Antenna optimization through space mapping. +IEEE +Transaction on Antennas and Propagation, 2007. + +Modeling Scattering Coefficients in Antenna Design using Self-Attentive Complex Polynomials with Image-based +Representation +A. Derivations +Theorem 3.1 (Analytical Structure of Scattering Coeffi- +cients). For ODE in the form of Eqn. 3, if A(h) is diago- +nalizable, then the logarithm of modulus of the scattering +coefficients log |S11(ω)| can be written as: +log |S11(ω)| = log |c0(h)| + +K +� +k=1 +log |ω − zk(h)| +|ω − pk(h)| +(4) +where the constant c0(h), zeros {zk(h)}K +k=1 and poles +{pk(h)}K +k=1 are complex numbers and all depend on A(h) +and thus functions of the design choice h. +Proof. Consider the following high-dimensional linear +ODE problem of N variables: +˙φ = Aφ +(7) +Here φ is a N-dimensional vector and we refer its com- +ponent at spatial location x as φ(x) (i.e. a scalar), and +A is an N-by-N diagonalizable matrix. A = A(h) de- +pends on material and topological properties, i.e., the de- +sign choice h, is time-invariant and is not necessarily sym- +metric. +Therefore, A has the following decomposition +A = UΛU −1, where each column of U is its eigenvector +and Λ = diag(λ1, . . . , λN) is a diagonal matrix containing +all its eigenvalues {λi}N +i=1. Note that entries of both U +and Λ can be complex numbers. Since the ODE is stable +and all excitation eventually vanishes, the real part of all +eigenvalues are negative: ℜ[λi] < 0. +By theory of linear ODE, we know that the solution to Eqn. 7 +has analytic form: +φ(t) = eAtφ(0) +(8) +where φ(0) is the initial condition of φ. Then we can +compute its (single-sided) Fourier transform ˆφ(ω) := +� +∞ +0 +φ(t)e−iωtdt: +ˆφ(ω) += +�� +∞ +0 +eAte−iωtdt +� +φ(0) += +U +�� +∞ +0 +eΛte−iωtdt +� +U −1φ(0) += +Udiag +�� +∞ +0 +eλite−iωtdt +� +U −1φ(0) += +Udiag +� +1 +iω − λi +� +U −1φ(0) +(9) +Note that U and φ(0) are all time-invariant. +For +double-sided +Fourier +transform +ˆφ(ω) +:= +� +∞ +−∞ φ(t)e−iωtdt +and +symmetric +signal +extension +φ(−t) = φ(t) = eA|t|φ(0) to negative side, similar we +can compute +� +∞ +−∞ +eλi|t|e−iωtdt += +1 +iω − λi ++ +1 +−iω − λi +(10) += +− +2λi +λ2 +i + ω2 +(11) +Note that while it looks like a real number, the result is still +complex since the eigenvalue λi is in general a complex +number for an asymmetric dynamical system A. In this +case, we can write ˆφ(ω) as: +ˆφ(ω) = −Udiag +� +2λ1 +λ2 +1 + ω2 , . . . , +2λN +λ2 +N + ω2 +� +U −1φ(0) +In both cases, each component of ˆφ(ω) is a rational function +of complex polynomial with respect to frequency ω. Since +all voltages and currents in the antenna are linear functions +of EM quantities represented in the components of φ, they +are also rational function of complex polynomials. As a +result, the input impedance Zin(ω) of the antenna, defined +as the ratio between the voltage and the current, is also a +rational function of complex polynomials, represented as a +quotient of two complex polynomials Q1(ω) and Q2(ω): +Zin(ω) = Q1(ω) +Q2(ω) +(12) +Note that this holds regardless of where the voltage and the +currents are defined. +Therefore, the scattering coefficient S11(ω) has the follow- +ing structure: +S11(ω) := Zin(ω)/Z0 − 1 +Zin(ω)/Z0 + 1 = Q1(ω) − Z0Q2(ω) +Q1(ω) + Z0Q2(ω) +(13) +Therefore, it can be represented as a ratio of two complex +polynomials of the same degrees (called it K). +According to the fundamental theorem of algebra, any poly- +nomial of order K can be written as a product of order-1 +factors ω − ωk and a constant, where {ωk} are the (com- +plex) roots of the K-th order polynomials. Therefore, the +log spectrum of S11(ω) is: +log |S11(ω)| = log |c0(h)| + +K +� +k=1 +log |ω − zk(h)| +|ω − pk(h)| +(14) +where the constant c0(h), zeros {zk(h)}K +k=1 and poles +{pk(h)}K +k=1 are all functions of A = A(h) and thus the +design choice h. +Remark. +Note that it is possible that the polynomial +Q1(ω) − Z0Q2(ω) may not have the same order as the + +Modeling Scattering Coefficients in Antenna Design using Self-Attentive Complex Polynomials with Image-based +Representation +polynomial Q1(ω) + Z0Q2(ω) (i.e., one of them has their +leading term precisely cancelled out, while the other does +not). While this is a rare situation, when it happens, Eqn. 14 +still applies, by having one or more zeros/poles moving far +away from the concerned frequency region of ω. Then the +corresponding factor |ω − zk| (or |ω − pk|) almost never +changes, and can be absorbed into the constant term c0. +B. Specification of the Design Space +The dimensions sk += (sk,x, sk,y) and ranges for the +location lk,x, lk,y of each of the 5 patches pk are +p1: s1 = (0.75, 5.49), l1,x ∈ [0, 10], l1,y ∈ [0.5, 0.5] +p2: s2 = (17.64, 1.7), l2,x ∈ [0, 12.36], l2,y ∈ [1, 4.7] +p3: s3 = (11.38, 3.0), l3,x ∈ [10, 18.62], l3,y ∈ [1, 3] +p4: s4 = (18.63, 0.56), l4,x ∈ [0, 11.37], l4,y ∈ [1, 5.44] +p5: s5 = (0.99, 2.43), l5,x ∈ [10, 29.01], l5,y ∈ [−2, 3.57] +where all values are in mm. These values were determined +from antennas that have been used for past production de- +vices in industry. Additionally, patch p1 determines the +location of the discrete port. +C. Experimental details +Hyperparameter +Value +Batch Size +100 +Learning Rate +.0005 +Activation +Swish +Warmup epochs +100 +Decay LR plateau epochs +20 +Decay LR plateau factor +.5 +Total Epochs +500 +Attention heads +8 +Attention layers +[2,4,6,8] +CZP Degree +[8,12,16,20] +Table 1. General experimental hyperparameters +Hyperparameter +Value +Spatial Attention Maps +16 +Conv KernelxStridexPad +5x1x2 +Conv Layers +2 +Conv Filters +128 +Attn dim feedforward +256 +Table 2. Image input specific hyperparameters for the transformer +with spatial attention +D. Other Visualizations +Hyperparameter +Value +FC embed dimension +256 +FC layers +2 +Attn dim feedforward +512 +Table 3. Coordinate input specific hyperparameters for the trans- +former with coordinate input + +Modeling Scattering Coefficients in Antenna Design using Self-Attentive Complex Polynomials with Image-based +Representation +(a) CZP +(b) Raw +Figure 8. Comparison of predicted frequency response (a) CZP and (b) Raw with an 8 layer transformer and image input on two test set +examples, with low MSE (top row) and high MSE (bottom row). Red is the frequency response predicted by the surrogate model, green is +the frequency response from CST. For easy examples, CZP and raw are both smooth. For hard examples, CZP is smooth by design but +raw may be non-smooth. + +prediction +-0 +*********** +simulation +-2 +-4: +(dB) +-6 : +-8 - +-10 : +2 +3 +0 +4 +5 +6 +1 +Frequency (GHz)prediction +-0 +米米 +simulation +-2 : +-4 +B +p) +S +-8 - +-10: +-12 +0 +3 +4 +5 +6 +7 +Frequency (GHz)prediction +0 +simulation +-2 : +-4 +B +p) +-6 +5 +-8 . +-10 +-12 +0 +2 +3 +4 +5 +6 +7 +Frequency (GHz)prediction +-0 +simulation +-2 +-4 - +B +p) +5 +XIXIX +X +-8 - +-10: +-12 +7 +0 +L +2 +3 +4 +5 +6 +Frequency (GHz)Modeling Scattering Coefficients in Antenna Design using Self-Attentive Complex Polynomials with Image-based +Representation +Figure 9. A random antenna configuration (above line) and four +(of sixteen total) attention maps (below line) learned by the spa- +tial attention component of the proposed transformer architecture +(overlayed on the antenna configuration). The intensity of the red +shading indicates the activations of the attention map. Activations +are greatest around boundaries and corners. + diff --git a/c9E0T4oBgHgl3EQf5AL3/content/tmp_files/load_file.txt b/c9E0T4oBgHgl3EQf5AL3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..053d44c3acc0d9a9ef32dffa0487e05ca4ba5de0 --- /dev/null +++ b/c9E0T4oBgHgl3EQf5AL3/content/tmp_files/load_file.txt @@ -0,0 +1,745 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQf5AL3/content/2301.02747v1.pdf,len=744 +page_content='Modeling Scattering Coefficients in Antenna Design using Self-Attentive Complex Polynomials with Image-based Representation Andrew Cohen∗ 1 Weiping Dou 1 Jiang Zhu 1 Slawomir Koziel 2 Peter Renner 1 Jan-Ove Mattsson 1 Xiaomeng Yang 1 Beidi Chen 1 Kevin Stone 1 Yuandong Tian∗ 1 Abstract Finding antenna designs that satisfy frequency requirements and are also optimal with respect to multiple physical criteria is a critical compo- nent in designing next generation hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQf5AL3/content/2301.02747v1.pdf'} +page_content=' How- ever, such a process is non-trivial because the objective function is typically highly nonlinear and sensitive to subtle design change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQf5AL3/content/2301.02747v1.pdf'} +page_content=' Moreover, the objective to be optimized often involves elec- tromagnetic (EM) simulations, which is slow and expensive with commercial simulation software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQf5AL3/content/2301.02747v1.pdf'} +page_content=' In this work, we propose a sample-efficient and accurate surrogate model, named CZP (Constant Zeros Poles), to directly estimate the scattering coefficients in the frequency domain of a given 2D planar antenna design, without using a simu- lator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQf5AL3/content/2301.02747v1.pdf'} +page_content=' CZP achieves this by predicting the com- plex zeros and poles for the frequency response of scattering coefficients, which we have theo- retically justified for any linear PDE, including Maxwell’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQf5AL3/content/2301.02747v1.pdf'} +page_content=' Moreover, instead of using low-dimensional representations, CZP leverages a novel image-based representation for antenna topology inspired by the existing mesh-based EM simulation techniques, and attention-based neural network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQf5AL3/content/2301.02747v1.pdf'} +page_content=' We demonstrate experi- mentally that CZP not only outperforms baselines in terms of test loss, but also is able to find 2D an- tenna designs verifiable by commercial software with only 40k training samples, when coupling with advanced sequential search techniques like reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQf5AL3/content/2301.02747v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQf5AL3/content/2301.02747v1.pdf'} +page_content=' Introduction The next generation of Metaverse computing devices such as virtual reality (VR) and augmented reality (AR) offers Equal contribution 1Meta AI 2Reykjavic University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQf5AL3/content/2301.02747v1.pdf'} +page_content=' Corre- spondence to: Andrew Cohen 1. This assumption +is based on observation that populations of excited levels are largely +controlled by spontaneous deexcitation and the time scale of this pro- +cess is much smaller than kinematic timescale of ∇ · (v𝑛𝑖). The first +level is mostly controlled by spontaneous recombinations with much +larger timescale. We also assume that populations of excited levels +are much smaller than electron concentration 𝑛𝑒 and first level pop- +ulation 𝑛1. Thus, the system of equation and the constraint equation +(2) becomes +���� +���� +𝜎1 += ∇ · (v𝑛1) +𝜎𝑖 += 0 for 𝑖 > 1 +𝑛𝑒 += 𝑛H − 𝑛1. +(7) +The initial conditions are obtained by solving the equations +𝜎𝑖 = 0, 𝑖 ≥ 1 +at the beginning of each stream line. +The line profile is computed using ray-by-ray integration of ra- +diation transfer equation (Muzerolle et al. 2001). The absorption +coefficient is computed using Doppler profile, and full frequency +redistribution is assumed. Here another important parameter arises: +angle 𝑖 between line of sight and the axis of the magnetosphere. +Because the observed profile is weak (see Fig. 1), it is impossible +to separate magnetosphere component from the spectrum of the pho- +tosphere with sufficient precision. So, for each ray that passes through +the surface of the star we put the photosphere spectrum 𝐼★𝜈 , shifted +to correct for the solid-body rotation of the star, in the equation for +ray intensity +𝐼𝜈 = 𝐼★ +𝜈 𝑒−𝜏𝜈 + +𝜏𝜈 +∫ +0 +𝑆𝜈𝑒𝜏𝑑𝜏. +(8) +MNRAS 000, 1–7 (2015) + +4 +D. V. Dmitriev et al. +Table 1. Parameter grid. +Parameter +Minimum value +Step +Maximum value +Units +log �𝑀 +-8.4 +0.2 +-11 +M⊙/yr +𝑇max +7000 +1000 +15000 +K +𝑅in +2 +1 +10 +𝑅★ +𝑊 +1 +0.2 +4 +𝑅★ +𝑖 +35 +5 +60 +Degrees +Here 𝜏𝜈 is optical depth along the line of sight and 𝑆𝜈 is the source +function. +Subsequently, five parameters are needed to compute the line pro- +file, excluding the parameters of the star: accretion rate �𝑀, maximum +temperature in the magnetosphere 𝑇max, inner radius of the magne- +tosphere 𝑅in, width of the magnetosphere 𝑊 and angle 𝑖. +4 FITTING PROCEDURE AND RESULTS +We used the grid of the parameters with a total number of 108864 +points described in Table 1. Because the magnetospheric accretion is +impossible at the distances larger then the corotation radius 𝑅cor, the +models where outer radius 𝑅out = 𝑅in + 𝑊 exceeded 11𝑅★ were dis- +regarded, leaving a total number of 86184 computed profiles. This +estimate of the corotation radius was calculated assuming that the +rotational velocity on the equator 𝑣eq is equal to 𝑣 sin𝑖 = 12 km s−1 +obtained from the spectrum, because the inclination angle 𝑖 is un- +known. The true value of corotation radius 𝑅cor must be ≤ 11 R★. +For each obtained profile the residual with observations 𝛿 was +computed +𝛿 = +� +� +� +� +� +1 +𝑁freq +𝑁freq +∑︁ +|𝑣 |≥30 km s−1 +(𝑟mod(𝜈) − 𝑟obs(𝜈))2, +(9) +where 𝑟mod = 𝐼mod/𝐼𝑐 is the computed profile, 𝑟obs = 𝐼obs/𝐼𝑐 is the +observed profile and 𝑁freq is the number of frequencies where the +profile was computed. The central part of the profile where |𝑣| = +𝑐|𝜈 − 𝜈H𝛼|/𝜈H𝛼 > 30 km s−1 is removed from the sum because our +model cannot produce strong enough emission at this region. Similar +to Thanathibodee et al. (2020), we add a central Gaussian component +𝑟𝜈 = 𝑟mod + 𝐴 exp +� +−𝑐2(𝜈 − 𝜈H𝛼)2 +2𝜈2 +H𝛼Δ𝑣2 +� +, +(10) +to remove this difference. Its parameters 𝐴, Δ𝑣 are computed by +fitting 𝑟mod(𝜈)−𝑟obs(𝜈). This central component may originate in an +accretion shock at the base of the magnetosphere (Dodin 2015, 2018) +or in active regions in the chromosphere of the star. The existence +of these regions is supported by the X-ray activity of RZ Psc (Punzi +et al. 2018). We emphasize again, that this narrow central component +was omitted in the computation of residuals. +Fig. 3 shows an example of theoretical profile constructed from the +different components in comparison with observed one. We found, +that magnetosphere accretion model with weak central peak can +explain virtually the entirety of the observed profile. The only dis- +crepancy worth of discussion is positioned at 𝑣 ≈ −110 km s−1 as +can be seen in Fig. 3. This velocity coincides with one of the BACs +observed in NaI lines (see Fig. 1), so this may be an H𝛼 absorption +from the same outflowing gas. +We derive observed profile parameters by computing mean of pa- +rameters of the models with sufficiently low residuals 𝛿 < +√ +2𝛿min, +where 𝛿min is minimum value of 𝛿 on the grid. For error estimate +Figure +3. +One +of +the +computed +H𝛼 +profiles +(red +line +labeled +mag+center+phot) with 𝛿 < +√ +2𝛿min in comparison with observations +(black line labeled obs). Magnetosphere only profile (without photosphere, +yellow line labeled mag), central component (red dashed line labeled center) +and deviation from observation (gray dashed line labeled mod-obs) are also +shown. The model parameters are: �𝑀 = 2.5 × 10−10 M⊙/yr, 𝑇max = 9000 +K, 𝑅in = 5 R★, 𝑊 = 2 R★, 𝑖 = 45◦. The central component has 𝐴 = 0.12 +and Δ𝑣 = 16 km/s. The significant deviation from the observed profile at +𝑣 ≈ −110km s−1 is marked. +we use standard deviation +√ +𝐷 where 𝐷 is the dispersion of such pa- +rameters. However, its important to highlight the limitations of this +approach. If, for example, for one of the parameters the inequality +𝛿 < +√ +2𝛿min is true for all grid points, this approach will yield the +midpoint as the result and the square root of the dispersion of the +grid points √︁𝐷grid as an error. Both those values are completely +independent of observations, so, in reality, this parameter is unde- +termined. To highlight such situations we compute the confidence +relation √︁𝐷grid/ +√ +𝐷 for each of the parameters. If this relation is +close to unity the parameter is considered unconstrained. In our case +such situation occurs for two parameters of the magnetosphere: width +𝑊 and maximum temperature 𝑇max. We chose ranges for these pa- +rameters that are in agreement with the results of theoretical works +(see Hartmann et al. (1994), Muzerolle et al. (2001), Lima et al. +(2010)). +We use one of NaI optical doublet component (5890 Å) and one +of CaII infrared triplet component (8542 Å) to restrict the model +parameters. In the RZ Psc spectrum observed at November 16 2013 +there is no noticeable absorption in the red wing of NaI 5890 Å, but +there is a profound absorption feature in CaII 8542Å from ∼50 to +∼500 km s−1 very similar to absorption feature observed in H𝛼 (see +Fig. 1). We argue that this is due to the fact that the magnetosphere +is optically thin in NaI 5890 Å, but optically thick in CaII 8542Å, +and we can use this fact to disregard some of the models with small +𝛿. To achieve this we computed absorption coefficients in these lines +for the conditions arising in the magnetosphere using Cloudy (Fer- +land et al. 2017) package and calculated absorption profiles of the +magnetosphere with 𝛿 < +√ +2𝛿min at 200 km/s. Then we rejected the +models for which this value was smaller than 0.9𝐼𝑐 for NaI 5890 Å +or bigger than 0.9𝐼𝑐 for CaII 8542 Å. +To illustrate the effect of advective term ∇ · (v𝑛1) in equations (7) +we also calculated profiles in the stationary assumption (𝜎𝑖 = 0, see +section 3). Fig. 4 shows how the residual 𝛿 depends on accretion rate +�𝑀 and temperature 𝑇max in the stationary assumption. Fig. 5 is the +same, but with the advective term taken into account. The minimum +residual value 𝛿min is approximately 0.017 for both cases. It can be +MNRAS 000, 1–7 (2015) + +Magnetospheric accretion onto RZ Psc +5 +o +Figure 4. Minimum residual value 𝛿 for computed models in the stationary +case with fixed �𝑀 and 𝑇max. The residual value is shown in color. The region +where 𝛿 < +√ +2𝛿min and absorption in CaII and NaI lines satisfies our criteria +lies inside the red border. The dashed red border separates models where +𝛿 < +√ +2𝛿min but the criteria is not satisfied. +o +Figure 5. Minimum residual value 𝛿 for computed models with fixed �𝑀 and +𝑇max with advective term ∇ · (𝑣𝑛1) taken into account. The residual value is +shown in color. The region where 𝛿 < +√ +2𝛿min and absorption in CaII and +NaI lines meets our criteria lies inside the red border. The dashed red border +separates models where 𝛿 < +√ +2𝛿min but the criteria is not satisfied. +seen clearly that stationary assumption produces smaller accretion +rates for low temperatures, although the difference is only about half +an order. This agrees with theoretical results described in Dmitriev +& Grinin (2022). This difference allows us to disregard temperatures +smaller than 9000 K using CaII and NaI lines (in stationary assump- +tions these temperatures are valid, see Fig. 4). Subsequently, this has +a significant impact on the precision of obtained accretion rate, as for +low temperatures accretion rates up to 10−9 M⊙yr−1 are required to +produce observed H𝛼 profile. +The obtained average values of parameters are presented in Ta- +ble 2. The accretion rate �𝑀, inner radius of the magnetosphere 𝑅in +and inclination angle 𝑖 are well determined with confidence relation +√︁𝐷grid/ +√ +𝐷 ≈ 3. However, for temperature 𝑇max and magnetosphere +width 𝑊 this relation is close to 1. This is due to the fact that those +two parameters are only bounded from bellow on the grid, as can be +seen in Fig. 5. However, the width 𝑊 cannot be much larger, because +the outer radius 𝑅out = 𝑅in + 𝑊 cannot exceed the corotation radius. +Table 2. Results with advection taken into account. Parameters in red rows +are unconstrained due to their low confidence relation. +Parameter +Value +Error +Confidence +Units +log �𝑀 +-10.1 +±0.3 +3.0 +M⊙/yr +𝑇max +12500 +±2100 +1.4 +K +𝑅in +5.5 +±0.9 +3.0 +R∗ +𝑊 +3.0 +±0.6 +1.9 +R∗ +𝑖 +43 +±3 +3.8 +Degrees +5 DISCUSSION +According to results of our modeling the logarithm of mass accretion +rate onto RZ Psc during the "flare" of its accretion activity on 2013 +Nov. 16 was log �𝑀 = −10.1 ± 0.3 ( �𝑀 ≈ 7 × 10−11M⊙yr−1) that +is, about 10 times more than before the burst. This suggests that the +accretion process at the late stages of the Pre-Main Sequence evolu- +tion is extremely unsteady. But even at the moment of the maximal +accretion activity the accretion rate onto RZ Psc was very small com- +pared to typical rates of T Tauri stars 10−8 − 10−7 M⊙ yr−1. This is +probably one of the reasons for sustained accretion activity in ≈20 +Myr old RZ Psc system. The other reason for existence of the long +living disk around RZ Psc is the operation of accretion process in the +weak magnetic propeller mode (Grinin et al. 2015). It explains the +very interesting property of this star outside of rare accretion bursts: +existence of the spectroscopic signatures of the matter outflow and +lack of any signs of accretion. In the paper cited above we argued +that the terminal velocity of the expelled gas does not exceed the +local escape velocity. Romanova et al. (2018) called such a mode +of accretion as the "soft" propeller. In this case the magnetosphere +works as a mixer. It is a very economical mode of accretion when the +CS gas is expelled from the star and return back into the disk. Such +a disk can survive during a very long time. +The weak accretion rate in the RZ Psc system indicates, that the +ionization in the falling gas can deviate from equilibrium. In the case +of RZ Psc accounting of this effect allows us to reject models with +low temperature using CaII and NaI lines and determine the accretion +rate and other parameters more precisely. This result demonstrates +importance of the temperature diagnostic for the modeling of mag- +netospheres in young stars). +Using the obtained values of +�𝑀 and 𝑅in one can estimate the +strength of the dipole component of the magnetic field on the equator +of RZ Psc 𝐵dip. Assuming that 𝑅in is the truncation radius we can +rewrite equation (2.2) from Bouvier et al. (2007) as +𝐵dip = +� +𝑅in +7.1 R★ +�7/4 � +�𝑀 +10−8 M⊙/yr +�1/2 � +𝑀★ +0.5 M⊙ +�1/4 � 𝑅★ +2 R⊙ +�−5/4 +kGs. +(11) +Substituting values of �𝑀 and 𝑅in from Table 2 we obtain +𝐵dip = +� 5.5 +7.1 +�7/4 � 10−10.1 +10−8 +�1/2 � 1.1 +0.5 +�1/4 � 1.2 +2 +�−5/4 +≈ 0.13±0.08 kGs. +This value is significantly lower than the typical value (𝐵 ≈ 1 kGs) +observed for T Tauri stars. +From the Table 2 we have outer radius of the magnetosphere +𝑅out = 𝑅in +𝑊 ≈ 8.5±1.5 𝑅★. According to (Grinin et al. 2015) the +corotation radius of RZ Psc is about 8-9 𝑅★. This value coincides +with our estimation of the outer radius of the magnetosphere 𝑅out. +Therefore our estimate of the magnetic field admits existence of +matter outflow in the magnetic propeller mode from the outer regions +of the magnetosphere. This explains the presence of both accretion +and outflow signatures in the spectrum (see Fig. 1). But, if we put +MNRAS 000, 1–7 (2015) + +6 +D. V. Dmitriev et al. +the accretion rate observed outside the accretion burst +�𝑀 = 7 × +10−12 M⊙yr−1 and the estimated magnetic field (≈ 0.1 kGs) in the +equation (11) then the inner radius of the magnetosphere will extend +to ≈ 10 𝑅★. This value is larger then the corotation radius, which +explains the absence of accretion signatures and existence of only +outflow signatures in the spectra obtained in the normal state of the +star. +In our calculations we used the classical model of the stellar mag- +netosphere based on the dipole magnetic field. The recent observa- +tions of magnetic fields in the WTTS’s demonstrate the large diversity +in strengths and topology of the large-scale magnetic field (Donati +et al. 2011, 2014, 2017; Hill et al. 2017, 2019; Nicholson et al. +2018; Yu et al. 2017). In the light of this the direct measurements of +magnetic field in RZ Psc are highly desirable. +From the point of view of the variable CS extinction model, the +inclination angle 𝑖 is one of the key parameters of CS disks. Our +modeling showed that the inclination angle of RZ Psc 𝑖 = 43 ± 3◦. +This value is smaller in comparison with the inclination angle 𝑖 ≈ 70◦ +of the photometrically active UXOrs (Kreplin et al. 2013, 2016; +Pontoppidan et al. 2007; Langlois et al. 2018), and this difference is +probably the main reason of the low photometric variability of RZ +Psc. +6 CONCLUSIONS +In this paper we modeled H𝛼 emission in the spectrum of RZ Psc +during the accretion burst in November 2013 using a magnetosphere +model described in Dmitriev et al. (2019) and Dmitriev & Grinin +(2022). The main results can be summarized as follows: +(i) The accretion rate increased approximately by an order of +magitude to the value of log �𝑀 = −10.1 ± 0.3, that corresponds to +�𝑀 = 7 × 10−11M⊙yr−1. Outside the episode of the accretion burst, +the accretion rate is too small to produce any noticable accretion +signatures. +(ii) The inclination angle 𝑖 = 43 ± 3◦ is low compared to the +typical one for UX Ori stars 𝑖 ≈ 70◦, which can be a reason of the +low photometric variability of the star. +(iii) The accounting for advective effects allowed us to place a +lower limit on the temperature in the magnetosphere at ≈ 10000 K +using observed profiles of the IR CaII triplet lines and D Na I reso- +nance lines, which significantly improved precision of our estimate +of accretion rate. +(iv) The magnetosphere extends approximately to the corotation +radius. Thus, at the outermost regions the magnetic field can ex- +pel some of the accreting gas. This explains the presence of BACs +attributed to the magnetic propeller in the November 16 spectrum +observed during the accretion burst. +(v) We estimate the dipole magnetic field component as 𝐵dip ≈ +0.1 kGs using obtained values of accretion rate and inner radius of +the magnetosphere. This value is quite low for T Tauri stars. In this +regard it would be interesting to directly measure the magnetic field +of RZ Psc. +ACKNOWLEDGEMENTS +The authors thank the referee for useful suggestions that helped +to improve the manuscript. DVD, TAE and VPG acknowledge +the support of Ministry of Science and Higher Education of +the Russian Federation under the grant no. 075-15-2020-780 +(N13.1902.21.0039). This research has made use of the Keck +Observatory Archive (KOA), which is operated by the W. M. +Keck Observatory and the NASA Exoplanet Science Institute +(NExScI), under contract with the National Aeronautics and Space +Administration. +DATA AVAILABILITY +All data used in this article will be shared on reasonable request to +the corresponding author. +REFERENCES +Bouvier J., Alencar S. H. P., Harries T. J., Johns-Krull C. M., Romanova +M. M., 2007, in Reipurth B., Jewitt D., Keil K., eds, Protostars and +Planets V. p. 479 (arXiv:astro-ph/0603498) +Dmitriev D. V., Grinin V. P., 2022, Astronomy Letters, 48, 29 +Dmitriev D. V., Grinin V. P., Katysheva N. A., 2019, Astronomy Letters, 45, +371 +Dodin A. V., 2015, Astronomy Letters, 41, 196 +Dodin A., 2018, MNRAS, 475, 4367 +Donati J. F., et al., 2011, MNRAS, 417, 472 +Donati J. F., et al., 2014, MNRAS, 444, 3220 +Donati J. F., et al., 2017, MNRAS, 465, 3343 +Ferland G. J., et al., 2017, Revista Mexicana de Astronomia y Astrofisica, 53, +385 +Grachev S. I., Grinin V. P., 1975, Astrophysics, 11, 20 +Grinin V. P., Kiselev N. 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N., Müller A., +Moerchen M., 2013, A&A, 553, L1 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–7 (2015) + diff --git a/dNFJT4oBgHgl3EQf_C3-/content/tmp_files/load_file.txt b/dNFJT4oBgHgl3EQf_C3-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e69983d26bf739cd6f2a4ccba56b706095f713e3 --- /dev/null +++ b/dNFJT4oBgHgl3EQf_C3-/content/tmp_files/load_file.txt @@ -0,0 +1,661 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf,len=660 +page_content='MNRAS 000, 1–7 (2015) Preprint 30 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='0 Magnetospheric accretion at the late phases of the Pre-Main-Sequence evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The case of RZ Psc D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Dmitriev, 1,2★ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Ermolaeva,1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Grinin1,3 and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Potravnov4,5 1Central (Pulkovo) Astronomical Observatory of the Russian Academy of Sciences, Pulkovskoye Chausse 65/1, 196140, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='Petersburg, Russia 2Crimean Astrophysical Observatory of the Russian Academy of Sciences, p/o Nauchny, 298409, Republic of Crimea 3St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Petersburg State University, Universitetskii pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 28, 198504, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Petersburg, Russia 4Institute of Solar-Terrestrial Physics, Siberian branch of Russian Academy of Sciences, Lermontov Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 126A, 664033, Irkutsk, Russia 5Institute of Astronomy of the Russian Academy of Sciences, Pyatnitskaya str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 48, 119017, Moscow, Russia Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' in original form ZZZ ABSTRACT It has been shown that during theburst of accretion activity observed in UX Ori type star RZ Psc in 2013, the accretion rate increased approximately by an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This means that the accretion process at the late stages of the Pre-Main Sequence evolution is very unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Using the spectra obtained during this episode we have studied the magnetospheric emission in the H𝛼 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Models of magnetospheric accretion are calculated to obtain the parameters of the magnetosphere from this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' In present work we have taken into account the influence of the recombination delay effect during gas motion in the stellar magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The accounting for this effect and the presence of the magnetospheric absorption in the IR CaII triplet lines and its absence in D Na I resonance lines allowed us to place a lower limit on the temperature in the magnetosphere at ≈ 10000 K, which significantly improved precision of our estimate of accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' According to the best fit model the logarithm of accretion rate is log �𝑀 = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='3 ( �𝑀 ≈ 7 × 10−11 M⊙yr−1) and the inclination angle of RZ Psc is 43 ± 3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' It is less than the inclination, typical for the UX Ori stars (about 70◦), that explains the weak photometric variability of this star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Using the obtained accretion rate and magnetosphere radius we estimate the strength of the dipole component of the magnetic field of RZ Psc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='1 kGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Key words: accretion, accretion disks – radiative transfer – stars: individual: RZ Psc – stars: pre-main-sequence – stars: magnetic fields 1 INTRODUCTION The star RZ Psc (Sp = K0 IV, Herbig (1960)) is one of the most un- usual members of the UX Ori stars (UXOrs) family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The members of this family are the photometrically most active young objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' They demonstrate the deep (up to 2-3𝑚 in V band) sporadic brightness minima with a typical duration from a few days to a few weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The reason for this activity is the complex structure of the nearest cir- cumstellar (CS) environment of young stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' When the direct stellar radiation is blocked by a CS dust cloud crossing the line of sight the scattered radiation of CS disk dominates, and UXOr becomes a highly polarized object (Grinin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (1991)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This important obser- vational fact tells us about the small inclination of CS disk planes of UX Ori stars relative to the line of sight as the main reason of their photometric activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Unlike typical UXORs the star RZ Psc shows very short Algol- like minima with the typical duration of 1-2 days (Zaitseva 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Pugach 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Such short eclipses are similar to those observed in the eclipsing binary systems, although many attempts to find a period were unsuccessful (Kennedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2017) and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' For a long time evolutionary status of the star was unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' RZ Psc is not located close to any known star formation regions: its galactic ★ E-mail: @.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (dmitrievdv242@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='com) lattitude is high (about 35 degrees) and there are no emission lines in star spectrum (Herbig 1960;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Kaminski˘ı et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' No infrared (IR) excess was observed in JHK bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Spectral observations by Herbig (1960) did not reveal any signatures of youth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The first signs of a dusty disk (or disk-like envelope) have been observed by Kiselev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' It was found that the linear polar- ization of the star increased up to approximately 5% during the deep minimum, that is typical for UXOrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' These observations were con- firmed by Shakhovskoi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The latter authors first suggested that RZ Psc is surrounded by circumstellar disk with the central cav- ity free (or almost free) of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This assumption was confirmed by de Wit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2013): a bright mid-IR excess was found in WISE observations of RZ Psc, fitted by black-body radiation with tempera- ture ≈ 500 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' They assumed that the star is surrounded by the dusty ring with inner radius 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='7 AU and this assumption was recently confirmed by Kennedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' They discovered that RZ Psc hosts a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='12 M⊙ companion at a projected separation of 23 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The spectroscopic observations have shown that the Li I 6708 Å line is present in the spectrum of RZ Psc and has an equivalent width EW(Li) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='202 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='010 Å (Grinin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Using the lithium depletion trend and kinematical treatment of RZ Psc the first age estimate of approximatly 30-40 Myr was made for the star, that was later refined by Potravnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2019) to 20+3 −5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This allowed us © 2015 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='11693v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='SR] 27 Jan 2023 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Dmitriev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' to reinforce the evolutionary status of RZ Psc as the post UX Ori star (Grinin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' So, in terms of accretion activity RZ Psc is a weak lined T Tauri star (WTTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' At the same time, according to de Wit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2013) the star has quite strong mid IR excess (𝐿IR ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='08𝐿bol) that is not typical for WTTS’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The first evidence for existence of a magnetosphere around RZ Psc came from observations of the narrow blue-shifted absorption com- ponents (BACs) in the sodium D NaI lines (Grinin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' To explain the origin of these components, the model of magnetospheric accretion in the magnetic propeller mode was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Estimates have shown that for realisation of this regime, the magnetosphere must be large, extending up to 10 stellar radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The spectroscopic monitor- ing of RZ Psc revealed occasional presence of the weak emission in the core of the photospheric H𝛼 line (Potravnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Inter- estingly, the H𝛼 emission was detected when the star was near the bright photometric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' An estimation of the mass accretion rate �𝑀 ≤ 7 × 10−12 M⊙yr−1 was made using the empirical calibration of the H𝛼 flux versus accretion luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The very interesting spectral observations of RZ Psc have been made by Punzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2018) at November 2013, after deep photomet- ric minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The H𝛼 line in that spectra demonstrated the classical signs of accretion: the red-shifted absorption component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The accre- tion rate estimated from this profile using the same empirical method was ≈ 5 × 10−11 M⊙yr−1 (Potravnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' So, Punzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2018) observed a burst of accretion activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' We assume that during the accretion burst a quasistable magneto- sphere was formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' That is true, if the duration of the burst 𝑡burst was sufficiently larger than the free fall time from the base of the magne- tosphere (𝑡ff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The fact, that the redshifted absorption component is observed at large interval of the velocities (from approximately 100 km/s to almost the escape velocity of the RZ Psc ≈ 600 km/s) means that the gas have managed to fall onto the star before the burst ended, and thus 𝑡burst ≥ 𝑡ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The regular look of the absorption component and the fact, that we were able to reproduce the observed profile supports this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The goal of our paper is to model the H𝛼 line profile to determine parameters of the magnetosphere: temperature, accretion rate, size and inclination angle of the magnetosphere axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The model we are using is described in detail in our previous papers (Dmitriev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Dmitriev & Grinin 2022) and based on the classical approach to modeling of T Tauri stars magnetospheres (Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Muzerolle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' However, unlike previously developed mod- els, our model takes into account advective transfer of ionization that can be important in low density plasma when collisional recombina- tion and ionization processes are slow (see section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' As it is shown in Dmitriev & Grinin (2022), this effect can be important at the low accretion rates �𝑀 ≤ 10−9 M⊙yr−1, like in the case of RZ Psc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The description of observational data used is given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Section 3 gives a brief review of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The results and details of fitting procedure are given in Section 4 and discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' A short review of main results is given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2 OBSERVATIONAL DATA In our analysis we used high resolution optical spectrum retrieved from the Keck Observatory Archive1, demonstrative for the accretion activity of RZ Psc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This is essentially the same material which was 1 https://www2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='keck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='edu/koa/public/koa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='php −110 −54 −54 −31 −31 −31 −17 −17 −17 −110 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' H𝛼, CaII and NaI lines observed at November 16, 2013 in RZ Psc spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The BACs, noticeable in Ca ii 8542 Å and Na i 5889 Å profiles, are labeled on the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' previously discussed in Punzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Potravnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' We recall that accretion is almost ceased in ∼ 20 Myr old RZ Psc system and manifested as the short-term "accretion flares" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' It is a dif- ficult observational challenge to catch the star during such a sporadic accretion event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Hence, this high-quality spectrogram obtained at the period of enhanced accretion is still relevant and has not been sur- passed for revealing physical properties of accretion/outflow activity in RZ Psc system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The spectrogram was obtained on the night of November 16, 2013 with the Keck I telescope and HIRES echelle spectrograph (PI: B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='Zuckerman).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Observations were carried out with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='148 arc- sec projected slit width resulted in nominal spectral resolution 𝑅 ≈ 38000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The wavelength coverage of the spectrograms was Δ𝜆 ≈ 4700 − 9000 Å with the signal-to-noise ratio of about 𝑆/𝑁 ≈ 120 (per pixel) in the region near H𝛼 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Details on the data processing are given in Potravnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The processing work- flow including standard calibration routine for the science frames, 1D spectrum extraction and wavelength calibration were made with the Makee2 software (written by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='Barlow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The heliocentric corrections was applied to the wavelength scale and afterwards it was shifted for the the stellar radial velocity 𝑅𝑉 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='3 km s−1 (Potravnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Thus, hereafter we used the spectrum in the rest frame associated with the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The flux normalization was performed using the approximation of the continuum level with the low-order cubic spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 16 spectrogram was obtained soon after the deep mini- mum when the star was close to its normal brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The H𝛼 and Ca ii lines demonstrated clear inverse P Cyg profile, with filled-in photospheric component and broad redshifted absorption extended up to +580 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' BACs in sodium lines are clearly seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 1 and also traceable in profiles of Ca ii lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' No corresponding wind features were observed in the H𝛼 line (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 3 MODEL In our modeling we use the following parameters of RZ Psc: 𝑅★ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='2 R⊙, 𝑀★ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='1 M⊙ (Potravnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2019) and 𝑇★ = 5350 K (Potravnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Here only a brief description of the model 2 https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='edu/ tb/makee/ MNRAS 000, 1–7 (2015) Magnetospheric accretion onto RZ Psc 3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The schematic of the magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' is given, where the focus was shifted on the details most relevant for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The more detailed description can be found in our previous works Dmitriev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2019) and Dmitriev & Grinin (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' In the framework of the classical approach, the magnetosphere is assumed to be formed by the dipole field aligned with the stellar rotation where the gas falls freely along magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' In the scope of these assumptions it has 5 independent parameters: accretion rate �𝑀, maximum temperature in the magnetosphere 𝑇max, inner radius 𝑅in, width 𝑊 = 𝑅out−𝑅in, where 𝑅out is the outer radius of the magnetosphere and the initial velocity 𝑣start which should be close to the thermal velocity 𝑣th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Following Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (1994) and Muzerolle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2001) we set 𝑣start to 10 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' From these parameters density, temperature and motion of the gas are completely determined (see, for example, Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (1994)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2 the schematic of the magnetosphere is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' We assume purely hydrogen gas and write state equations in the form ∇ · (v𝑛𝑖) = 𝜎𝑖 for 𝑖 ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (1) with constraint equation 𝑛H = 𝑛𝑒 + ∞ ∑︁ 𝑖=1 𝑛𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2) where 𝑛𝑖 are level populations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 𝑛𝑒 is electron concentration and 𝑛H is total hydrogen concentration and 𝜎𝑖 are sources and sinks for 𝑖-th hydrogen level: 𝜎𝑖 = ∞ ∑︁ 𝑘=𝑖+1 𝑛𝑘 (𝐴𝑘𝑖 + 𝐵𝑘𝑖𝐽𝑘𝑖) + 𝑖−1 ∑︁ 𝑗=1 𝑛 𝑗 𝐵 𝑗𝑖𝐽𝑖 𝑗 + 𝑛𝑒 ∞ ∑︁ 𝑗≠𝑖 𝑛 𝑗𝑞 𝑗𝑖− 𝑛𝑖 ������ 𝑖−1 ∑︁ 𝑗=1 (𝐴𝑖 𝑗 + 𝐵𝑖 𝑗𝐽𝑖 𝑗) + ∞ ∑︁ 𝑘=𝑖+1 𝐵𝑖𝑘𝐽𝑖𝑘 + 𝑛𝑒 ∞ ∑︁ 𝑗≠𝑖 𝑞𝑖 𝑗 ������ + 𝑛2 𝑒(𝐶𝑖 + 𝐵𝑐𝑖) + 𝑛3 𝑒𝑄𝑐𝑖 − 𝑛𝑖𝐵𝑖𝑐 − 𝑛𝑖𝑛𝑒𝑞𝑖𝑐,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 𝑖 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (3) The terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (3) include radiative transitions (Einstein’s coeffi- cients 𝐴𝑖 𝑗, 𝐵𝑖 𝑗), transitions induced by collisions with free electrons (𝑞𝑖 𝑗), spontaneous and induced by radiation or electron collisions recombinations (𝐶𝑖, 𝐵𝑐𝑖 and 𝑄𝑐𝑖) and radiative and collisional ion- izations (𝐵𝑖𝑐 and 𝑞𝑖𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The sums are truncated at 15-th level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The mean intensities 𝐽𝑖 𝑗 are computed using the Sobolev’s approximation (Sobolev 1960;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Grachev & Grinin 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Rybicki & Hummer 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The radiative ionization and induced recombination coefficients (𝐵𝑖𝑐 and 𝐵𝑐𝑖) are computed as follows 𝐵𝑖𝑐 = 4𝜋 ∞ ∫ 𝜈𝑐𝑖 𝛼𝑖𝑐(𝜈) 𝐽𝜈 ℎ𝜈 𝑑𝜈 (4) 𝐵𝑐𝑖 = 𝑖2ℎ3 (2𝜋𝑚𝑒𝑘𝐵𝑇)3/2 𝑒 ℎ𝜈𝑐𝑖 𝑘𝐵𝑇 4𝜋 ∞ ∫ 𝜈𝑐𝑖 𝛼𝑖𝑐(𝜈) 𝐽𝜈 ℎ𝜈 𝑒− ℎ𝜈 𝑘𝐵𝑇 𝑑𝜈, (5) where 𝛼𝑖𝑐 is photoionization cross section from level 𝑖, 𝜈𝑖𝑐 is the threshold frequency for level 𝑖, 𝑇 is the local temperature in the magnetosphere and 𝐽𝜈 is the mean intensity in the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' We neglect the continuum radiation from the magnetosphere, and assume that the external sources of radiation are the blackbody radiation of the star and the accretion spot at the base of the magnetosphere 𝐽𝜈 = 𝑊★𝐵𝜈(𝑇★) + 𝑊spot𝐵𝜈(𝑇spot), (6) where 𝐵𝜈 is the Planck’s law, 𝑊★ and 𝑊spot are the geometrical dilution factors for the star and for the accretion spot and 𝑇spot is the temperature of the accretion spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This temperature is computed from the assumption that all of the kinetic energy of the falling gas at the base of the magnetosphere is radiated away as blackbody radiation (Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' In present paper we take into account the advective term ∇ · (v𝑛𝑖) in equations (1), because it can be significant for low accretion rates, observed in RZ Psc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Its importance in T Tau stars magnetospheres was first stated by Martin (1996), and the impact on the emission spectrum was first considered by Dmitriev & Grinin (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' To simplify the system of partial differential equations (1) we ne- glect advective term ∇·(v𝑛𝑖) for excited levels𝑖 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This assumption is based on observation that populations of excited levels are largely controlled by spontaneous deexcitation and the time scale of this pro- cess is much smaller than kinematic timescale of ∇ · (v𝑛𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The first level is mostly controlled by spontaneous recombinations with much larger timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' We also assume that populations of excited levels are much smaller than electron concentration 𝑛𝑒 and first level pop- ulation 𝑛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Thus, the system of equation and the constraint equation (2) becomes ���� ���� 𝜎1 = ∇ · (v𝑛1) 𝜎𝑖 = 0 for 𝑖 > 1 𝑛𝑒 = 𝑛H − 𝑛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (7) The initial conditions are obtained by solving the equations 𝜎𝑖 = 0, 𝑖 ≥ 1 at the beginning of each stream line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The line profile is computed using ray-by-ray integration of ra- diation transfer equation (Muzerolle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The absorption coefficient is computed using Doppler profile, and full frequency redistribution is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Here another important parameter arises: angle 𝑖 between line of sight and the axis of the magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Because the observed profile is weak (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 1), it is impossible to separate magnetosphere component from the spectrum of the pho- tosphere with sufficient precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' So, for each ray that passes through the surface of the star we put the photosphere spectrum 𝐼★𝜈 , shifted to correct for the solid-body rotation of the star, in the equation for ray intensity 𝐼𝜈 = 𝐼★ 𝜈 𝑒−𝜏𝜈 + 𝜏𝜈 ∫ 0 𝑆𝜈𝑒𝜏𝑑𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (8) MNRAS 000, 1–7 (2015) 4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Dmitriev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Parameter grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Parameter Minimum value Step Maximum value Units log �𝑀 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='2 11 M⊙/yr 𝑇max 7000 1000 15000 K 𝑅in 2 1 10 𝑅★ 𝑊 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='2 4 𝑅★ 𝑖 35 5 60 Degrees Here 𝜏𝜈 is optical depth along the line of sight and 𝑆𝜈 is the source function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Subsequently, five parameters are needed to compute the line pro- file, excluding the parameters of the star: accretion rate �𝑀, maximum temperature in the magnetosphere 𝑇max, inner radius of the magne- tosphere 𝑅in, width of the magnetosphere 𝑊 and angle 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 4 FITTING PROCEDURE AND RESULTS We used the grid of the parameters with a total number of 108864 points described in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Because the magnetospheric accretion is impossible at the distances larger then the corotation radius 𝑅cor, the models where outer radius 𝑅out = 𝑅in + 𝑊 exceeded 11𝑅★ were dis- regarded, leaving a total number of 86184 computed profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This estimate of the corotation radius was calculated assuming that the rotational velocity on the equator 𝑣eq is equal to 𝑣 sin𝑖 = 12 km s−1 obtained from the spectrum, because the inclination angle 𝑖 is un- known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The true value of corotation radius 𝑅cor must be ≤ 11 R★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' For each obtained profile the residual with observations 𝛿 was computed 𝛿 = � � � � � 1 𝑁freq 𝑁freq ∑︁ |𝑣 |≥30 km s−1 (𝑟mod(𝜈) − 𝑟obs(𝜈))2, (9) where 𝑟mod = 𝐼mod/𝐼𝑐 is the computed profile, 𝑟obs = 𝐼obs/𝐼𝑐 is the observed profile and 𝑁freq is the number of frequencies where the profile was computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The central part of the profile where |𝑣| = 𝑐|𝜈 − 𝜈H𝛼|/𝜈H𝛼 > 30 km s−1 is removed from the sum because our model cannot produce strong enough emission at this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Similar to Thanathibodee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2020), we add a central Gaussian component 𝑟𝜈 = 𝑟mod + 𝐴 exp � −𝑐2(𝜈 − 𝜈H𝛼)2 2𝜈2 H𝛼Δ𝑣2 � , (10) to remove this difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Its parameters 𝐴, Δ𝑣 are computed by fitting 𝑟mod(𝜈)−𝑟obs(𝜈).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This central component may originate in an accretion shock at the base of the magnetosphere (Dodin 2015, 2018) or in active regions in the chromosphere of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The existence of these regions is supported by the X-ray activity of RZ Psc (Punzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' We emphasize again, that this narrow central component was omitted in the computation of residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 3 shows an example of theoretical profile constructed from the different components in comparison with observed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' We found, that magnetosphere accretion model with weak central peak can explain virtually the entirety of the observed profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The only dis- crepancy worth of discussion is positioned at 𝑣 ≈ −110 km s−1 as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This velocity coincides with one of the BACs observed in NaI lines (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 1), so this may be an H𝛼 absorption from the same outflowing gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' We derive observed profile parameters by computing mean of pa- rameters of the models with sufficiently low residuals 𝛿 < √ 2𝛿min, where 𝛿min is minimum value of 𝛿 on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' For error estimate Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' One of the computed H𝛼 profiles (red line labeled mag+center+phot) with 𝛿 < √ 2𝛿min in comparison with observations (black line labeled obs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Magnetosphere only profile (without photosphere, yellow line labeled mag), central component (red dashed line labeled center) and deviation from observation (gray dashed line labeled mod-obs) are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The model parameters are: �𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='5 × 10−10 M⊙/yr, 𝑇max = 9000 K, 𝑅in = 5 R★, 𝑊 = 2 R★, 𝑖 = 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The central component has 𝐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='12 and Δ𝑣 = 16 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The significant deviation from the observed profile at 𝑣 ≈ −110km s−1 is marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' we use standard deviation √ 𝐷 where 𝐷 is the dispersion of such pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' However, its important to highlight the limitations of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' If, for example, for one of the parameters the inequality 𝛿 < √ 2𝛿min is true for all grid points, this approach will yield the midpoint as the result and the square root of the dispersion of the grid points √︁𝐷grid as an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Both those values are completely independent of observations, so, in reality, this parameter is unde- termined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' To highlight such situations we compute the confidence relation √︁𝐷grid/ √ 𝐷 for each of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' If this relation is close to unity the parameter is considered unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' In our case such situation occurs for two parameters of the magnetosphere: width 𝑊 and maximum temperature 𝑇max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' We chose ranges for these pa- rameters that are in agreement with the results of theoretical works (see Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (1994), Muzerolle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2001), Lima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' We use one of NaI optical doublet component (5890 Å) and one of CaII infrared triplet component (8542 Å) to restrict the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' In the RZ Psc spectrum observed at November 16 2013 there is no noticeable absorption in the red wing of NaI 5890 Å, but there is a profound absorption feature in CaII 8542Å from ∼50 to ∼500 km s−1 very similar to absorption feature observed in H𝛼 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' We argue that this is due to the fact that the magnetosphere is optically thin in NaI 5890 Å, but optically thick in CaII 8542Å, and we can use this fact to disregard some of the models with small 𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' To achieve this we computed absorption coefficients in these lines for the conditions arising in the magnetosphere using Cloudy (Fer- land et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2017) package and calculated absorption profiles of the magnetosphere with 𝛿 < √ 2𝛿min at 200 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Then we rejected the models for which this value was smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='9𝐼𝑐 for NaI 5890 Å or bigger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='9𝐼𝑐 for CaII 8542 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' To illustrate the effect of advective term ∇ · (v𝑛1) in equations (7) we also calculated profiles in the stationary assumption (𝜎𝑖 = 0, see section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 4 shows how the residual 𝛿 depends on accretion rate �𝑀 and temperature 𝑇max in the stationary assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 5 is the same, but with the advective term taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The minimum residual value 𝛿min is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='017 for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' It can be MNRAS 000, 1–7 (2015) Magnetospheric accretion onto RZ Psc 5 o Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Minimum residual value 𝛿 for computed models in the stationary case with fixed �𝑀 and 𝑇max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The residual value is shown in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The region where 𝛿 < √ 2𝛿min and absorption in CaII and NaI lines satisfies our criteria lies inside the red border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The dashed red border separates models where 𝛿 < √ 2𝛿min but the criteria is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' o Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Minimum residual value 𝛿 for computed models with fixed �𝑀 and 𝑇max with advective term ∇ · (𝑣𝑛1) taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The residual value is shown in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The region where 𝛿 < √ 2𝛿min and absorption in CaII and NaI lines meets our criteria lies inside the red border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The dashed red border separates models where 𝛿 < √ 2𝛿min but the criteria is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' seen clearly that stationary assumption produces smaller accretion rates for low temperatures, although the difference is only about half an order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This agrees with theoretical results described in Dmitriev & Grinin (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This difference allows us to disregard temperatures smaller than 9000 K using CaII and NaI lines (in stationary assump- tions these temperatures are valid, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Subsequently, this has a significant impact on the precision of obtained accretion rate, as for low temperatures accretion rates up to 10−9 M⊙yr−1 are required to produce observed H𝛼 profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The obtained average values of parameters are presented in Ta- ble 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The accretion rate �𝑀, inner radius of the magnetosphere 𝑅in and inclination angle 𝑖 are well determined with confidence relation √︁𝐷grid/ √ 𝐷 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' However, for temperature 𝑇max and magnetosphere width 𝑊 this relation is close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This is due to the fact that those two parameters are only bounded from bellow on the grid, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' However, the width 𝑊 cannot be much larger, because the outer radius 𝑅out = 𝑅in + 𝑊 cannot exceed the corotation radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Results with advection taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Parameters in red rows are unconstrained due to their low confidence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Parameter Value Error Confidence Units log �𝑀 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='1 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='0 M⊙/yr 𝑇max 12500 ±2100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='4 K 𝑅in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='5 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='0 R∗ 𝑊 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='0 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='9 R∗ 𝑖 43 ±3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='8 Degrees 5 DISCUSSION According to results of our modeling the logarithm of mass accretion rate onto RZ Psc during the "flare" of its accretion activity on 2013 Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 16 was log �𝑀 = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='3 ( �𝑀 ≈ 7 × 10−11M⊙yr−1) that is, about 10 times more than before the burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This suggests that the accretion process at the late stages of the Pre-Main Sequence evolu- tion is extremely unsteady.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' But even at the moment of the maximal accretion activity the accretion rate onto RZ Psc was very small com- pared to typical rates of T Tauri stars 10−8 − 10−7 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This is probably one of the reasons for sustained accretion activity in ≈20 Myr old RZ Psc system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The other reason for existence of the long living disk around RZ Psc is the operation of accretion process in the weak magnetic propeller mode (Grinin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' It explains the very interesting property of this star outside of rare accretion bursts: existence of the spectroscopic signatures of the matter outflow and lack of any signs of accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' In the paper cited above we argued that the terminal velocity of the expelled gas does not exceed the local escape velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Romanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2018) called such a mode of accretion as the "soft" propeller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' In this case the magnetosphere works as a mixer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' It is a very economical mode of accretion when the CS gas is expelled from the star and return back into the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Such a disk can survive during a very long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The weak accretion rate in the RZ Psc system indicates, that the ionization in the falling gas can deviate from equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' In the case of RZ Psc accounting of this effect allows us to reject models with low temperature using CaII and NaI lines and determine the accretion rate and other parameters more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This result demonstrates importance of the temperature diagnostic for the modeling of mag- netospheres in young stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Using the obtained values of �𝑀 and 𝑅in one can estimate the strength of the dipole component of the magnetic field on the equator of RZ Psc 𝐵dip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Assuming that 𝑅in is the truncation radius we can rewrite equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='2) from Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2007) as 𝐵dip = � 𝑅in 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='1 R★ �7/4 � �𝑀 10−8 M⊙/yr �1/2 � 𝑀★ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='5 M⊙ �1/4 � 𝑅★ 2 R⊙ �−5/4 kGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (11) Substituting values of �𝑀 and 𝑅in from Table 2 we obtain 𝐵dip = � 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='1 �7/4 � 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='1 10−8 �1/2 � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='5 �1/4 � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='2 2 �−5/4 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='13±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='08 kGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This value is significantly lower than the typical value (𝐵 ≈ 1 kGs) observed for T Tauri stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' From the Table 2 we have outer radius of the magnetosphere 𝑅out = 𝑅in +𝑊 ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='5 𝑅★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' According to (Grinin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2015) the corotation radius of RZ Psc is about 8-9 𝑅★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This value coincides with our estimation of the outer radius of the magnetosphere 𝑅out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Therefore our estimate of the magnetic field admits existence of matter outflow in the magnetic propeller mode from the outer regions of the magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This explains the presence of both accretion and outflow signatures in the spectrum (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' But, if we put MNRAS 000, 1–7 (2015) 6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Dmitriev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' the accretion rate observed outside the accretion burst �𝑀 = 7 × 10−12 M⊙yr−1 and the estimated magnetic field (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='1 kGs) in the equation (11) then the inner radius of the magnetosphere will extend to ≈ 10 𝑅★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This value is larger then the corotation radius, which explains the absence of accretion signatures and existence of only outflow signatures in the spectra obtained in the normal state of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' In our calculations we used the classical model of the stellar mag- netosphere based on the dipole magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The recent observa- tions of magnetic fields in the WTTS’s demonstrate the large diversity in strengths and topology of the large-scale magnetic field (Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2011, 2014, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2017, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' In the light of this the direct measurements of magnetic field in RZ Psc are highly desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' From the point of view of the variable CS extinction model, the inclination angle 𝑖 is one of the key parameters of CS disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Our modeling showed that the inclination angle of RZ Psc 𝑖 = 43 ± 3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This value is smaller in comparison with the inclination angle 𝑖 ≈ 70◦ of the photometrically active UXOrs (Kreplin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2013, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Pontoppidan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Langlois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 2018), and this difference is probably the main reason of the low photometric variability of RZ Psc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 6 CONCLUSIONS In this paper we modeled H𝛼 emission in the spectrum of RZ Psc during the accretion burst in November 2013 using a magnetosphere model described in Dmitriev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (2019) and Dmitriev & Grinin (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' The main results can be summarized as follows: (i) The accretion rate increased approximately by an order of magitude to the value of log �𝑀 = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='3, that corresponds to �𝑀 = 7 × 10−11M⊙yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Outside the episode of the accretion burst, the accretion rate is too small to produce any noticable accretion signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (ii) The inclination angle 𝑖 = 43 ± 3◦ is low compared to the typical one for UX Ori stars 𝑖 ≈ 70◦, which can be a reason of the low photometric variability of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (iii) The accounting for advective effects allowed us to place a lower limit on the temperature in the magnetosphere at ≈ 10000 K using observed profiles of the IR CaII triplet lines and D Na I reso- nance lines, which significantly improved precision of our estimate of accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (iv) The magnetosphere extends approximately to the corotation radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' Thus, at the outermost regions the magnetic field can ex- pel some of the accreting gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This explains the presence of BACs attributed to the magnetic propeller in the November 16 spectrum observed during the accretion burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' (v) We estimate the dipole magnetic field component as 𝐵dip ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='1 kGs using obtained values of accretion rate and inner radius of the magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This value is quite low for T Tauri stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' In this regard it would be interesting to directly measure the magnetic field of RZ Psc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors thank the referee for useful suggestions that helped to improve the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' DVD, TAE and VPG acknowledge the support of Ministry of Science and Higher Education of the Russian Federation under the grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' 075-15-2020-780 (N13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content='0039).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' This research has made use of the Keck Observatory Archive (KOA), which is operated by the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFJT4oBgHgl3EQf_C3-/content/2301.11693v1.pdf'} +page_content=' M.' metadata={'source': 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+{sajjadur,hannah,dan_z,estevam,eser}@megagon.ai +Abstract +Human-centered AI workflows involve stakeholders with multiple roles interacting +with each other and automated agents to accomplish diverse tasks. In this paper, +we call for a holistic view when designing support mechanisms, such as interaction +paradigms, interfaces, and systems, for these multifaceted workflows. +1 +Introduction +Human-centered AI (HCAI) research focuses on various aspects: analyzing practitioners’ work +practices along dimensions such as collaboration [6, 29], explainability [21], and trust [14]; conceptu- +alizing specific workflows to inform design guidelines [11, 13, 16, 17, 24]; and developing supporting +tools and frameworks [1, 20, 28]. HCAI workflows are multi-faceted, wherein stakeholders with +different roles — e.g., product managers, subject matter experts, and data scientists — perform +diverse tasks in various phases using different tools. Multiple roles introduce challenges related to +collaboration among stakeholders and interaction with tools. The iterative workflows necessitate +fine-grained provenance management [20]. The scope and diversity of domains (e.g., NLP, ML, +Vision) introduce challenges related to the modeling of domain semantics. To this end, we propose a +holistic design that employs plastic interfaces to enable seamless interactions, integrated workflows +for ease of task completion, and a graph-based system for managing workflows and provenance. +Figure 1: Multifaceted Human-centered AI (M-HCAI) conceptual diagram. +2 +M-HCAI Challenges +Figure 1 depicts the conceptual diagram of M-HCAI with various workflow components and system +requirements. A workflow is iterative with transitions among phases (a). However, a fine-grained +characterization, proposed by Rahman et al. [17], reveals the iterative nature of a phase with stake- +holders performing diverse tasks (c). Stakeholders may interact with different tools and interfaces +depending on their task requirement, familiarity, and expertise (d) [20, 27]. Stakeholders with differ- +ent roles collaborate in various phases within the workflow both synchronously (in-person and virtual +meetings) and asynchronously (e.g., Email, Slack, Google Docs) [17, 19, 29]. +36th Conference on Neural Information Processing Systems (NeurIPS 2022). +arXiv:2301.03656v1 [cs.DB] 9 Jan 2023 + +Workflow and Interactions +System +(c) Task Interfaces +(a) Phases +Data +Data +Modeling +Preparation +Jjupyter +用 +(b) Task Model +1 +Act +Interaction +Model +Modeling +Interact +4 +Building +Collaborate +Verify +Roles: humans +Ideate +and agents +Model +Evaluation +Provenance +Data +Content +View +Assess +Scientist +Manager +management +Model +Product +Workflow +Manager +Deployment +Management +(d) RolesSuch complexity of an MHCAI workflow introduces several challenges. First, iteration: iterative +workflows often force the users to switch multiple tools tediously, often within a single phase [3, 17, +20]. Second, interface rigidity: existing tools lack the plasticity to accommodate diverse roles and +expertise as stakeholders ideate and deliberate [5, 14]. Third, non-standardized collaboration: as +stakeholders collaborate, there may be conflict and misunderstanding [25, 26]. However, there is +a lack of standardization around documentation to record these discussions [10, 19, 29]. Finally, +domain diversity: when we factor in diverse domains within the HCAI setting such as text [4, 20], +image, graphs, and tables [15], the lack of formal abstractions and grammar makes it difficult to +define and instrument operations in the interfaces systematically. +Another aspect lacking from the HCAI discourse is the management of complicated workflows, which +necessitate robust system design. Zhang et al. [29] identified lack of provenance as a contributing +factor in obfuscation and loss of knowledge when data science teams share data. Rahman et +al. [17] advocate for instrumenting provenance management mechanisms as built-in features of +systems. Kandogan et al. [7] highlight how existing systems fail to connect and exploit context across +stakeholders and agents due to a lack of initiative in modeling data, people, and interactions. +3 +Multifaceted Human-centered AI: A Holistic View +We posit that the workflow- and system-level challenges are interdependent. Capturing the provenance +of interactions and decisions within the workflow requires provenance-aware interfaces with built-in +logging capabilities. For the underlying system to effectively persist and catalog the provenance +information, the interactions defined on the interfaces need grounding on sound principles. Such +principled design requires (a) building abstractions to capture domain semantics (e.g., texts contain +words, sentences, POS tags, and opinions), (b) defining operators that represent interactions over +those abstractions in an interface (similar to the grammar of interaction graphics [22]), and (c) explicit +modeling of stakeholder roles. Finally, managing these human-centered iterative workflows requires +developing frameworks that enable the integration of tools and enhance interface scalability [2, 18]. +In response, we propose the following design considerations for M-HCAI: +An integrated workflow supporting multiple paradigms. Avoiding the context switching overhead +during iteration requires integrating different modalities: programming environments and interactive +interfaces (e.g., spreadsheets and visualization tools.) Computational notebooks are suitable for such +integration where programming can be complemented by introducing interactive widgets [8, 28]. +However, notebooks are not suitable for non-programmers involved in the HCAI process. Moreover, +these tools neither manage the fine-grained provenance of user interactions nor support versioning. +Therefore, research efforts for bridging these gaps are crucial for the holistic M-HCAI system design. +Plastic interfaces as the basis for interaction. The multi-role inclusion challenge with computa- +tional notebooks can be addressed by developing custom interfaces that may act as boundary objects +for the stakeholders. Boundary objects are artifacts that are “both plastic enough to adapt to local +needs and the constraints of the several parties employing them, yet robust enough to maintain a +common identity across site” [9, 23]. Recent work introduces cross-platform capabilities to transform +interactive widgets in notebooks into web dashboards that enable shared understanding among stake- +holders within an organizational workflow [1]. However, these widgets are not provenance-aware, +and the corresponding interactions are domain semantics agnostic. Therefore, further augmentation +is required to employ these interfaces to operate in concert with the underlying system. +Systems for supporting M-HCAI. Interactions among automated agents and stakeholders of dif- +ferent expertise and roles may occur at any phase. Therefore, the context (where and when a task +or interaction occurred), scope (specific data domain and phase), and abstractions of interactions +will vary depending on the scenario. To this end, knowledge graphs [12] (KGs) may serve as the +core data model for the support systems managing M-HCAI workflows. KGs can explicitly capture +diverse stakeholder roles and automated agents, different interaction types, and the corresponding +context [7]. Existing work define grammars to capture operations for text [4, 20], and tabular data +analysis [15]. Capturing domain-specific interaction contexts such as operations and data types would +require defining such abstractions for M-HCAI. +In this position paper, we draw attention to the multi-faceted nature of HCAI while identifying +workflow and system-level challenges. We propose a research agenda that requires multi-disciplinary +research efforts spanning databases, AI, visualization, and HCI. +2 + +References +[1] A. Bäuerle, Á. A. Cabrera, F. Hohman, M. Maher, D. Koski, X. Suau, T. Barik, and D. Moritz. +Symphony: Composing interactive interfaces for machine learning. In CHI Conference on +Human Factors in Computing Systems, pages 1–14, 2022. +[2] M. Bendre, T. Wattanawaroon, S. Rahman, K. Mack, Y. Liu, S. Zhu, Y. Lu, P.-J. Yang, X. 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Proceedings of the ACM on Human-Computer Interaction, 4(CSCW1): +1–23, 2020. +4 + diff --git a/eNE2T4oBgHgl3EQfGQbA/content/tmp_files/load_file.txt b/eNE2T4oBgHgl3EQfGQbA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..663db61f0643bfffb5478fbb8ec292bd01dad402 --- /dev/null +++ b/eNE2T4oBgHgl3EQfGQbA/content/tmp_files/load_file.txt @@ -0,0 +1,364 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf,len=363 +page_content='Towards Multifaceted Human-Centered AI Sajjadur Rahman, Hannah Kim, Dan Zhang, Estevam Hruschka, Eser Kandogan Megagon Labs {sajjadur,hannah,dan_z,estevam,eser}@megagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content='ai Abstract Human-centered AI workflows involve stakeholders with multiple roles interacting with each other and automated agents to accomplish diverse tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' In this paper, we call for a holistic view when designing support mechanisms, such as interaction paradigms, interfaces, and systems, for these multifaceted workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' 1 Introduction Human-centered AI (HCAI) research focuses on various aspects: analyzing practitioners’ work practices along dimensions such as collaboration [6, 29], explainability [21], and trust [14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' conceptu- alizing specific workflows to inform design guidelines [11, 13, 16, 17, 24];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' and developing supporting tools and frameworks [1, 20, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' HCAI workflows are multi-faceted, wherein stakeholders with different roles — e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=', product managers, subject matter experts, and data scientists — perform diverse tasks in various phases using different tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Multiple roles introduce challenges related to collaboration among stakeholders and interaction with tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' The iterative workflows necessitate fine-grained provenance management [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' The scope and diversity of domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=', NLP, ML, Vision) introduce challenges related to the modeling of domain semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' To this end, we propose a holistic design that employs plastic interfaces to enable seamless interactions, integrated workflows for ease of task completion, and a graph-based system for managing workflows and provenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Figure 1: Multifaceted Human-centered AI (M-HCAI) conceptual diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' 2 M-HCAI Challenges Figure 1 depicts the conceptual diagram of M-HCAI with various workflow components and system requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' A workflow is iterative with transitions among phases (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' However, a fine-grained characterization, proposed by Rahman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' [17], reveals the iterative nature of a phase with stake- holders performing diverse tasks (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Stakeholders may interact with different tools and interfaces depending on their task requirement, familiarity, and expertise (d) [20, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Stakeholders with differ- ent roles collaborate in various phases within the workflow both synchronously (in-person and virtual meetings) and asynchronously (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=', Email, Slack, Google Docs) [17, 19, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' 36th Conference on Neural Information Processing Systems (NeurIPS 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content='03656v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content='DB] 9 Jan 2023 Workflow and Interactions System (c) Task Interfaces (a) Phases Data Data Modeling Preparation Jjupyter 用 (b) Task Model 1 Act Interaction Model Modeling Interact 4 Building Collaborate Verify Roles: humans Ideate and agents Model Evaluation Provenance Data Content View Assess Scientist Manager management Model Product Workflow Manager Deployment Management (d) RolesSuch complexity of an MHCAI workflow introduces several challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' First, iteration: iterative workflows often force the users to switch multiple tools tediously, often within a single phase [3, 17, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Second, interface rigidity: existing tools lack the plasticity to accommodate diverse roles and expertise as stakeholders ideate and deliberate [5, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Third, non-standardized collaboration: as stakeholders collaborate, there may be conflict and misunderstanding [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' However, there is a lack of standardization around documentation to record these discussions [10, 19, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Finally, domain diversity: when we factor in diverse domains within the HCAI setting such as text [4, 20], image, graphs, and tables [15], the lack of formal abstractions and grammar makes it difficult to define and instrument operations in the interfaces systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Another aspect lacking from the HCAI discourse is the management of complicated workflows, which necessitate robust system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' [29] identified lack of provenance as a contributing factor in obfuscation and loss of knowledge when data science teams share data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Rahman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' [17] advocate for instrumenting provenance management mechanisms as built-in features of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Kandogan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' [7] highlight how existing systems fail to connect and exploit context across stakeholders and agents due to a lack of initiative in modeling data, people, and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' 3 Multifaceted Human-centered AI: A Holistic View We posit that the workflow- and system-level challenges are interdependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Capturing the provenance of interactions and decisions within the workflow requires provenance-aware interfaces with built-in logging capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' For the underlying system to effectively persist and catalog the provenance information, the interactions defined on the interfaces need grounding on sound principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Such principled design requires (a) building abstractions to capture domain semantics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=', texts contain words, sentences, POS tags, and opinions), (b) defining operators that represent interactions over those abstractions in an interface (similar to the grammar of interaction graphics [22]), and (c) explicit modeling of stakeholder roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Finally, managing these human-centered iterative workflows requires developing frameworks that enable the integration of tools and enhance interface scalability [2, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' In response, we propose the following design considerations for M-HCAI: An integrated workflow supporting multiple paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Avoiding the context switching overhead during iteration requires integrating different modalities: programming environments and interactive interfaces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=', spreadsheets and visualization tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=') Computational notebooks are suitable for such integration where programming can be complemented by introducing interactive widgets [8, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' However, notebooks are not suitable for non-programmers involved in the HCAI process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Moreover, these tools neither manage the fine-grained provenance of user interactions nor support versioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Therefore, research efforts for bridging these gaps are crucial for the holistic M-HCAI system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Plastic interfaces as the basis for interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' The multi-role inclusion challenge with computa- tional notebooks can be addressed by developing custom interfaces that may act as boundary objects for the stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Boundary objects are artifacts that are “both plastic enough to adapt to local needs and the constraints of the several parties employing them, yet robust enough to maintain a common identity across site” [9, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Recent work introduces cross-platform capabilities to transform interactive widgets in notebooks into web dashboards that enable shared understanding among stake- holders within an organizational workflow [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' However, these widgets are not provenance-aware, and the corresponding interactions are domain semantics agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Therefore, further augmentation is required to employ these interfaces to operate in concert with the underlying system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Systems for supporting M-HCAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Interactions among automated agents and stakeholders of dif- ferent expertise and roles may occur at any phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Therefore, the context (where and when a task or interaction occurred), scope (specific data domain and phase), and abstractions of interactions will vary depending on the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' To this end, knowledge graphs [12] (KGs) may serve as the core data model for the support systems managing M-HCAI workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' KGs can explicitly capture diverse stakeholder roles and automated agents, different interaction types, and the corresponding context [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Existing work define grammars to capture operations for text [4, 20], and tabular data analysis [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Capturing domain-specific interaction contexts such as operations and data types would require defining such abstractions for M-HCAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' In this position paper, we draw attention to the multi-faceted nature of HCAI while identifying workflow and system-level challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' We propose a research agenda that requires multi-disciplinary research efforts spanning databases, AI, visualization, and HCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' 2 References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Bäuerle, Á.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Cabrera, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' How do data science workers collaborate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' roles, workflows, and tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' Proceedings of the ACM on Human-Computer Interaction, 4(CSCW1): 1–23, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} +page_content=' 4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfGQbA/content/2301.03656v1.pdf'} diff --git a/edAzT4oBgHgl3EQfoP0g/content/tmp_files/2301.01592v1.pdf.txt b/edAzT4oBgHgl3EQfoP0g/content/tmp_files/2301.01592v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..70f387ebefb57b08c6b571b94bf77d2f5c871d7b --- /dev/null +++ b/edAzT4oBgHgl3EQfoP0g/content/tmp_files/2301.01592v1.pdf.txt @@ -0,0 +1,1680 @@ +CarFi: Rider Localization Using Wi-Fi CSI +Sirajum Munir∗† +Bosch Research and Technology +Center +Pittsburgh, PA, USA +sirajum.munir@us.bosch.com +Hongkai Chen∗ +Shan Lin +Stony Brook University +Stony Brook, NY, USA +{hongkai.chen,Shan.X.Lin}@stonybrook.edu +Shiwei Fang∗ +Mahathir Monjur +Shahriar Nirjon +UNC Chapel Hill +Chapel Hill, NC, USA +{shiwei,mahathir,nirjon}@cs.unc.edu +ABSTRACT +With the rise of hailing services, people are increasingly relying on +shared mobility (e.g., Uber, Lyft) drivers to pick up for transporta- +tion. However, such drivers and riders have difficulties finding each +other in urban areas as GPS signals get blocked by skyscrapers, +in crowded environments (e.g., in stadiums, airports, and bars), at +night, and in bad weather. It wastes their time, creates a bad user ex- +perience, and causes more CO2 emissions due to idle driving. In this +work, we explore the potential of Wi-Fi to help drivers to determine +the street side of the riders. Our proposed system is called CarFi that +uses Wi-Fi CSI from two antennas placed inside a moving vehicle +and a data-driven technique to determine the street side of the rider. +By collecting real-world data in realistic and challenging settings +by blocking the signal with other people and other parked cars, we +see that CarFi is 95.44% accurate in rider-side determination in both +line of sight (LoS) and non-line of sight (nLoS) conditions, and can +be run on an embedded GPU in real-time. +1 +INTRODUCTION +As people rely on ride-hailing services, e.g., Uber and Lyft, it be- +comes increasingly important for drivers and riders to find each +other without a hitch. Currently, drivers and riders use smartphones, +which rely on GPS or cellular signals, to locate each other while +far apart, and require them to recognize each other while nearby. +However, in urban cities and areas like downtown, where there are +numerous skyscrapers, GPS signals often do not work. In addition, +there are places, e.g., in airports, where the drivers need to come +indoors (such as parking garages) to pick up riders where the build- +ing structure blocks GPS signals. Also, it is challenging to locate +the actual rider among many people in crowded environments like +stadiums, airports, theatres, and bars. Moreover, the situation can +worsen due to lack of visibility, e.g., at night and during bad weather +(such as rain, storm, and snow). This issue wastes the time of the +riders and drivers, causes more CO2 emissions due to idle driving, +causes frustration, and creates a bad user experience. +A recent Uber study shows that every user agreed that they do +not like to negotiate the pickup point, and 11 out of 16 users find +it hard to give directions to the driver when the user is at a new +place [1]. Based on our discussion with a few drivers and riders, +we find that determining the street side of the rider is very crucial. +This is because, if the car is on the other side of the street, the rider +sometimes must cross the street, which can be unsafe. Also, the +drivers do not want to make a U-turn and realize that they were on +∗Equal contribution. +†Corresponding author. +the right side in the first place, which leads to a double U-turn. So, +we focus on determining the street side of the riders. +Several solutions have been proposed to improve the rider pick- +up experience. For example, the vehicle can use a camera and facial +recognition [35] to identify the rider and subsequently compute the +location. However, facial recognition requires the rider to upload +his or her photo, which can be privacy-invasive. Moreover, for facial +recognition to work, the rider needs to be within the camera’s field +of view and occupy enough pixels to be successfully recognized +and have good lighting conditions. One can also ask the user to +scan the surroundings with his or her phone, and then a server can +perform 3D reconstruction [17] and matching [22] to the previously +established real-world model to compute the exact location of the +rider. However, this is a computation-intensive approach, and this +method also requires the world to be digitized and constructed to +allow such matching. As commercial products, Uber and Lyft have +multicolored LED-based lights for riders to recognize their cars. +However, such a solution does not work in broad daylight, and it is +a rider-oriented solution, i.e., the rider has to find the car, and the +driver does not have much information about the location/side of +the rider. Our proposed solution overcomes these limitations. +In this paper, we perform an exploratory study to understand +the feasibility of using Wi-Fi to address this problem. We propose +to utilize smartphones – which the riders will mostly like to possess +for the ride-hailing booking – and Wi-Fi-enabled dashcams – in- +creasingly crucial for safety and legal purposes – to determine the +street side of the rider. We call our proposed system CarFi. CarFi +neither requires the rider to upload any photos of him or her nor a +photo of the surrounding area, which protects the rider’s privacy, +reduces the computation load, and does not depend on lighting +conditions. To this end, CarFi uses Wi-Fi communications between +the rider’s smartphone and the dashcam onboard the vehicle. The +usage of the dashcam is for the purpose of standalone devices that +can be installed on any vehicle, but the proposed system does not +exclude vehicles that have Wi-Fi already installed to localize the +rider. +CarFi uses a two-antenna Wi-Fi chipset embedded in the dash- +cam to receive the Wi-Fi packets sent by the smartphone held by +the rider. This system does not require any modification to the +vehicle and the smartphone. The Wi-Fi packets can be generated +by the ride-hailing app, which can share the phone’s MAC address +(or, a randomized MAC address) through the cloud/server to the +vehicle (or, driver’s app). Thus, the vehicle can listen to the packets +generated from the target phone. The system on the vehicle extracts +the Channel State Information (CSI) data from the Wi-Fi chipset. +arXiv:2301.01592v1 [cs.NI] 21 Dec 2022 + +After some preprocessing, it performs sub-carrier selection. Then, it +extracts relevant features (amplitude difference between antennas, +multipath profile, power delay profile) for rider-side determination. +Then, the contextual and motion-related features are encoded into +a data-driven model (LSTM) to classify whether the rider is on the +right or the left side of the vehicle. Our approach only uses CSI +amplitude and does not use CSI phase information. Thus, it avoids +effort for phase calibration. +This work has the following contributions: +• First, we perform a comprehensive exploratory analysis to +understand the potential of using Wi-Fi CSI in an automotive +environment for shared mobility applications. Our empiri- +cal study involves determining the set of features that can +effectively work in an automotive environment in both line +of sight (LoS) and non-line of sight (nLoS) conditions when +a vehicle is being driven and encoding the features into the +design and implementation of a data-driven model (LSTM) +for estimating the side of the rider using only two anten- +nas and CSI amplitude. Our CarFi system does not require +privacy-invasive personal information from the rider such as +a photo, avoids heavy computation on the server, and works +in the dark. +• Second, we set up an infrastructure to collect Wi-Fi CSI from +a moving vehicle with a drone-based system for annotating +the ground truth location of the vehicle when each packet is +received. We collect a dataset of 85 rides with over 568,000 +Wi-Fi packets in a realistic and challenging environment, +considering both LoS and nLoS, where other people and +other parked vehicles block Wi-Fi signals. To the best of our +knowledge, this is the first dataset to investigate how Wi-Fi +CSI changes over time when the receiver is placed inside a +moving car for shared mobility applications. +• Third, based on evaluation using data collected from the +real-world, our results show that CarFi is 95.44% accurate in +classifying the rider side in both LoS and nLoS conditions. +We also implement several baseline solutions using phase +difference and other features and show the superiority of our +solution. We also evaluate the execution time of our approach +in both powerful and embedded GPUs and show that our +solution can be run on an embedded GPU in real-time. +2 +CARFI OVERVIEW +An overview of the CarFi system is shown in Figure 1. When a +rider wants to travel to a specific location, s/he uses the ride-hailing +phone app on the phone to book a trip. The cloud server of the +service providers processes the request and finds a driver. The loca- +tions of the vehicle and the rider are determined by their respective +location providers, such as GPS on the phone. Once the trip is con- +firmed, the driver heads toward the rider’s location. As the driver +arrives within a certain distance, e.g., 0.5 miles from the rider based +on the location data, the rider’s phone will transmit Wi-Fi packets +at a higher transmission rate as the ride-hailing app controls it. In +the meantime, the phone’s MAC address is shared with the dashcam +via the servers in the cloud. A randomized temporary MAC address +can be used to preserve the privacy of the rider. As the vehicle +is also within this certain range, the dashcam starts listening for +Wi‐Fi Packets  +Filtering +Feature  +Extraction +Pre‐ +processing +Wi‐Fi  +Transmission +CSI  +Extraction +Rider’s  +Phone App +MAC  +Address +LSTM +Left vs. Right  +classification +Driver’s  +Phone App +Visualization  +of Rider’s side +Figure 1: CarFi system overview +Wi-Fi packets containing the phone’s MAC address and filters out +other packets. When CarFi system receives the Wi-Fi packets with +matched MAC address, it extracts the CSI information, performs +some pre-processing, and calculates relevant features. Then it feeds +the features to an LSTM, which estimates the street side of the rider. +Then, this information is passed to the driver’s smartphone app +from the dashcam for visualization. The data exchange between +the phone and the dashcam can be achieved via either Bluetooth or +cellular connection (if the dashcam has it). +3 +CHALLENGES +In this section, we discuss the challenges that CarFi system faces +for rider side localization in an automotive environment. +3.1 +Automotive Environment +When moving Wi-Fi devices from indoor locations to automotive +environments, the characteristics of the environment and its effects +on the signals change dramatically. One of the biggest issues in +an automotive environment is the metal structure of the vehicle +body, which can be similar to a Faraday cage. Although the signal +of normal radio frequency communication systems has a higher +frequency than what the window can block due to its large size, the +vehicle’s metal surface can still block and redistribute the signal. +Unfortunately, there has not been much work to understand how +Wi-Fi CSI looks like inside of a vehicle when the vehicle is being +driven. +With such a complex RF environment, the current state-of-the- +art method, such as SpotFi [18], can not accurately estimate the +Angle of Arrival (AoA) of the Wi-Fi signal. An example of such an +AoA estimation is shown in Figure 2(a). The X-axis represents the +distance of the rider from the car. The car is coming from the left +side of the X-axis, meets the rider in the center, and then leaves. +The three antenna arrays are placed at the center of the dashboard +of the car, and the AoA should be 0 to -90°(0 to 90°) when the rider is +at the right (left) side. We consider two cases: the rider is standing +without anyone blocking the signal (Figure 2(a)), and two other cars +and three other people blocking the signal (Figure 2(b)). The rider +was on the right side in both cases. We see that in LoS cases, the +AoA is relatively stable as the Wi-Fi signal penetrates through the +front windshield, but when the car leaves the rider, there is a lot of +fluctuation of the AoA as the backside of the car blocks the signal. +We observe that when other people and cars block the rider, the + +D20 +15 +10 +5 +0 +5 +10 +15 +20 +Rider distance from the car (m) +90 +75 +60 +45 +30 +15 +0 +15 +30 +45 +60 +75 +90 +AoA (degree) +AoA +(a) AoA estimation of when the rider is in LoS condition. +20 +15 +10 +5 +0 +5 +10 +15 +20 +Rider distance from the car (m) +90 +75 +60 +45 +30 +15 +0 +15 +30 +45 +60 +75 +90 +AoA (degree) +AoA +(b) AoA estimation of when the rider is in nLoS condition. +Figure 2: SpotFi AoA in LoS and in nLoS conditions. The rider is in the right side of the car. The expected AoA is 0 to -90°. +AoA is unreliable even when the rider is in front of the car. Since +AoA estimation also requires three antennas and phase calibration, +we do not use AoA in our approach. +3.2 +Speed and Time +We do not expect that the vehicle will approach the rider at highway +speed when they are nearby. Instead, we assume that the vehicle will +be traveling at a lower speed to be able to stop quickly. Therefore, +we assume 10 to 20 miles per hour vehicle speed, which translates +to 4.47 to 8.94 meters per second. We also consider the transmission +range of the Wi-Fi signal to be around 70 to 120 meters in the +outdoor environment. If the rider is 70 meters in front of the car, +the driver has about 7.83 to 15.66 seconds to stop the vehicle. Given +the human response time is about 1 to 1.5 seconds, we determine +that the if CarFi system takes 3 seconds, it will provide adequate +time for the driver to respond and stop safely. Smartphones can +transmit several hundreds of Wi-Fi packets in a second. However, +there could be a burst of packet loss due to non-line of sight (nLoS). +In addition, the more time we take to make a decision, the higher +accuracy we can offer. Thus, a small window size with a variable +number of received packets poses a difficult challenge for rider side +determination. +3.3 +Cost +In order to make the solution practical, we need to use inexpensive +antennas and a lightweight computing platform. A simpler solu- +tion might use two directional antennas to classify left vs. right. +However, we need directional antennas with 180-degree horizontal +beamwidth, which is expensive. For example, [2] costs $225 per +antenna. Cheaper ones have a smaller beamwidth. For example, +[3] costs $35.94 per antenna, but has only 66 degrees horizontal +beam patterns. Also, such directional antennas are bulky and could +obstruct the field of view of the driver more. Adding more antennas +also helps in improving the accuracy but also increases the cost of +the Wi-Fi chipset and antenna chain. Moreover, the solution needs +to be lightweight to be able to run on an embedded GPU or acceler- +ators. Although, such an accelerator would increase hardware cost, +a dashcam with such capability could provide additional benefits to +the drivers by offering additional services e.g., detecting accidents, +violence/aggression in the car and providing necessary support by +performing audio-visual analysis. +4 +APPROACH +In this section, we describe the CarFi approach in details. +4.1 +Pre-processing +When the receiving unit starts to receive Wi-Fi packets, CarFi +timestamps each packet and keeps all the packets within a window +size of 3 seconds for processing together. Then, it uses a stride +length of 0.4 seconds to create the next window. +4.2 +Feature selection +In this section, we discuss the set of features that we use for left vs. +right classification. +4.2.1 +Amplitude difference. We use Channel State Information +(CSI) from only two antennas for the classification. We assume +the distance between them is 𝑑. In our exploratory analysis, we +have 𝑑 = 5.2 cm. CSI contains how the RF signal propagates through +the environment as they are being affected during transmission. +The CSI data collected at the receiver side contains those affected +and encoded in the complex form with amplitude and phase infor- +mation. Each CSI data point is also the Channel Frequency Response +(CFR): +𝐻 (𝑓 ;𝑡) = +𝑁 +∑︁ +𝑛 +𝑎𝑛(𝑡)𝑒−𝑗2𝜋 𝑓 𝜏𝑛 (𝑡) +(1) +Where 𝑎𝑖 (𝑡) is the amplitude attenuation factor, 𝜏𝑖 (𝑡) is the prop- +agation delay, and 𝑓 is the carrier frequency [40] [24]. +Figure 3 shows how CSI amplitude difference between antenna +𝐶 and antenna 𝐴 looks like for a portion of a ride for 30 sub-carriers. +The rider was on the right side of the car. The X-axis shows the +distance of the car with respect to the rider. The car is approaching +from the left side of the X-axis, meets the rider at the middle of the +X-axis, and then passes the rider after that. When we plot amplitude +difference, we plot the CSI amplitude of the antenna 𝐶 - antenna 𝐴, +where antenna 𝐴, 𝐵, and 𝐶 are placed from left to right parallel to +the dashboard (Figure 8). So, a positive value is a good indicator that +the rider is on the right side. We see that the amplitude difference +values fluctuate over time, and they also vary for different sub- +carriers. While Figure3(a) shows a LoS condition, Figure 3(b) shows +a nLoS condition where the three other people and two other cars +were placed between the rider and the Wi-Fi receiver. We see a burst +of packet loss there. As the CSI amplitude varies by subcarriers, +instead of relying on all the sub-carriers, we determine the relevant +sub-carriers for us that are less prone to noise. +4.2.2 +Sub-carrier selection. Instead of relying on all the sub-carriers, +we select sub-carriers that are more resilient to noise. First, we + +(a) Amplitude difference of antennas (𝐶 - 𝐴) when the rider is in LoS. +(b) Amplitude difference of antennas (𝐶 - 𝐴) when the rider is in nLoS. +Figure 3: CSI amplitude difference in LoS and nLoS conditions. +Figure 4: CSI Amplitude difference between antennas (𝐶 - 𝐴) of 12 +selected sub-carriers using Variance-based Subcarrier Selection. +compute the covariance of CSI amplitude of all the subcarriers +of antenna C. High covariance between these subcarriers shows +they receive effective signal and not the noise. For each subcarrier +in antenna C, we select the corresponding subcarrier of antenna +A. These subcarriers have similar path properties (e.g., multipath +effect, attenuation) and receive correlated CSI data. We vary the +number of selected subcarriers from 1 to 30 and choose the number +of subcarriers that provide the highest accuracy. Please note that +sub-carriers are selected per window of packets. So, different win- +dows may have different sets of sub-carriers. We call this approach +Variance-based Sub-carrier Selection (VbSS). When choosing 𝑁 +subcarriers, we choose 𝑁 − 1 sub-carriers using VbSS and add the +first subcarrier. Figure 4 shows the amplitude difference of the 12 +selected subcarriers of a window when the rider is on the right side +of the car. +4.3 +Multipath Profile +Since Wi-Fi CSI data contains multipath attenuation caused by the +environment, the multipath profile extracted from the CSI data +can be very useful in location estimation. It can effectively pro- +vide whether the rider is in LoS or nLoS conditions. To extract the +multipath profile of the CSI data, we explore how MUSIC [36] and +SpotFi [18] algorithms extract signals and estimate their Angle of +Arrival. Inspired by this, we first isolate multiple possible signals +by performing Eigen decomposition of matrix 𝑋𝑋𝐻, where 𝑋 is +the CSI measurement, and 𝑋𝐻 is the conjugate transpose of 𝑋. The +eigenvectors and eigenvalues can be used as features as they are +affected by the environment and the vehicle. We take the top two +dominant multipaths and plot them in Figure 5 with both LoS and +nLoS conditions, and the rider was on the right side. The X-axis in +(a) First and second multipaths of when the rider is LoS condition. +(b) First and second multipaths of when the rider is nLoS condition. Three other +people and two other cars are blocking the signal. +Figure 5: Multipath profile in LoS and nLoS conditions. +both figures shows the distance from the car as the car is approach- +ing the rider from the left side of the axis. Figure 5(a) represents +the case when the rider is in a LoS condition. We see that as the +car approaches, there is a significant difference between the first +and the second multipath. However, as the car leaves the rider, the +backside of the car blocks the signal and causes nLoS conditions, +and hence the difference between the top two multipaths decreases +significantly. Figure 5(b) represents the case when the rider is in +a nLoS condition. There were three other people and two other +cars blocking the signal. As a result, the first and second dominant +multipath is closer from the beginning. However, when the car +passes the rider, it gets a line of sight for a brief moment, and hence +the difference between the first and the second multipath becomes +more prominent. We take the top two Eigenvalues representing the +top two dominant multipaths computed using CSI values of two +antennas as features for classification. +4.4 +Power Delay Profile +Power Delay Profile (PDP) describes the power level associated with +each multipath along with the propagation delays. However, due to +the limited bandwidth of the Wi-Fi channels, the path length resolu- +tion is not very precise. For our 802.11ac 40 MHz channel, the path +length resolution is 7.5m. But it can be helpful for coarse-grained +mobility tracking over time and provide contextual information +regarding LoS and nLoS. +When the Wi-Fi chipset measures the channel frequency re- +sponse as written in Equation 1, instead of measuring continuously, +it samples the response at discrete frequency points 𝑓 = 𝑓0 + 𝑘Δ𝑓 , +where k is the sub-carrier index and Δ𝑓 = 312.5𝑘𝐻𝑧 [49]. Since +Equation 1 is in the frequency domain, by applying Inverse Fourier +Transform, we can get the response in the time domain which is +also the Channel Impulse Response (CIR): +𝑓 (𝑡) = +𝑁 +∑︁ +𝑛 +𝑎𝑛𝛿(𝑡 − 𝜏𝑡) +(2) + +Difference +20 +Subcarrier: 1 +Subcarrier: 16 +Subcarrier: 2 +Subcarrier: 17 +Subcarrier: 3 +Subcarrier: 18 +10 +Subcarrier: 4 +Subcarrier: 19 +Subcarrier: 5 +Subcarrier: 20 +Amplitude +Subcarrier: 6 +Subcarrier: 21 +Subcarrier: 7 +Subcarrier: 22 +Subcarrier: 8 +Subcarrier: 23 +Subcarrier: 9 +Subcarrier: 24 +-10 +Subcarrier: 10 +Subcarrier: 25 +Subcarrier: 11 +Subcarrier: 26 +CSI +Subcarrier: 12 +Subcarrier: 27 +20 +15 +10 +-5 +0 +5 +10 +15 +20 +Subcarrier: 13 +Subcarrier: 28 +Riderdistancefromthecar(meters) +Subcarrier: 14 +Subcarrier: 29 +Subcarrier: 15 +Subcarrier: 30AmplitudeDifference +15 +Subcarrier: 1 +Subcarrier: 16 +10 +Subcarrier: 2 +Subcarrier: 17 +Subcarrier: 3 +Subcarrier: 18 +Subcarrier: 4 +Subcarrier: 19 +Subcarrier: 5 +Subcarrier: 20 +0 +Subcarrier: 6 +Subcarrier: 21 +Subcarrier: 7 +Subcarrier: 22 +Subcarrier: 8 +Subcarrier: 23 +Subcarrier: 9 +Subcarrier: 24 +10 +Subcarrier: 10 +Subcarrier: 25 +Subcarrier: 11 +Subcarrier: 26 +15 +Subcarrier: 12 +Subcarrier: 27 +20 +15 +10 +-5 +0 +5 +10 +15 +20 +Subcarrier: 13 +Subcarrier: 28 +Riderdistancefromthecar (meters) +Subcarrier: 14 +Subcarrier: 29 +Subcarrier: 15 +Subcarrier: 30CSIAmplitudeDifference +Subcarrier 13 +Subcarrier 4 +Subcarrier 5 +00 +Subcarrier 15 +Subcarrier 14 +Subcarrier 11 +Subcarrier 7 +Subcarrier3 +Subcarrier 8 +Subcarrier 19 +Subcarrier 20 +Subcarrier 1 +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +Packet numbers() +First multipath +15000 +Second multipath +12500 +10000 +7500 +5000 +2500 +0 +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +Rider distance from the car (meters)7000 +Firstmultipath +Second multipath +6000 +5000 +4000 +3000 +2000 +1000 +-5 +0 +5 +10 +Rider distance from the car (meters)(a) Power delay profile of when the rider is LoS condition.. +(b) Power delay profile of when the rider is nLoS condition. +Figure 6: Power delay profile in LoS and nLoS conditions. +where 𝑎𝑛 and 𝑁 is the same as in Equation 1 and 𝛿(·) is the +delta function. By calculating the norm ∥𝑓 (𝑡)∥2 of the Channel +Impulse Response 𝑓 (𝑡), we can get the Power Delay Profile. Each +of the signal samples in the Channel Impulse Response correlates +to different multipath as their time to travel from the transmitter +to the receiver differs due to differences in the traveled length. By +considering the IFFT theory, the time resolution Δ𝜏 is related to +the sampling resolution Δ𝑓 mentioned above. While increasing the +number of bins in IFFT, the actual resolution does not change. As +such, we set our IFFT bins to the number of subcarriers, which is +also the frequency sampling resolution. For our collected data, 30 +subcarriers are reported for each antenna. By using two antennas, +we obtain 60 PDP values as features per Wi-Fi packet. We show the +PDP values from one antenna in LoS condition in Figure 6(a), and +in nLoS condition in Figure 6(b), where there are three people and +two cars blocking the signal between the rider and his car. In both +cases, the rider was on the right side. We see how the PDP values +are changing as the car approaches the rider from the left side of +the X-axis and passes him. The PDP values do not necessarily tell +if the rider is on the left or right side but help contextualize the +packets of similar distance, and in LoS/nLoS conditions to provide +additional information to the classification model. +5 +CLASSIFICATION +We consider different classifiers to classify the side of the rider (left +vs. right), including k-Nearest Neighbor (kNN), Decision Tree (DT), +and Support Vector Machine (SVM). In addition, we design a Long +Short-Term Memory (LSTM) neural network classifier by effectively +integrating all the features. Now, we describe the design of an LSTM +and how we encode the relevant contextual and motion-related +features. +As the vehicle approaches the rider, the motion of the vehicle, +as well as the distance between the transmitter (the phone held by +the rider) and the receiver (Wi-Fi receiver on the vehicle), provides +additional features in the time domain. For example, Wi-Fi signal +differences between different antennas can vary across time. There +are also Wi-Fi signal differences on the same antenna with different +transmitter and receiver distances. Unlike neural network architec- +tures such as Fully-Connected Neural Network and Convolutional +Neural Network (CNN), LSTM can better encode time series data +with its feedback connections to remember values over arbitrary +time intervals. Thus it can exploit the temporal features introduced +by the vehicle’s motion. While traditional classifiers like k-NN, DT, +and SVM can capture features at a single time step, they lack the +ability to take into account the temporal features as the signal is +coming from either left or right in both LoS and nLoS situations. +The general execution of LSTM is described in equations below: +𝑖𝑡 = 𝜎(𝑊𝑖𝑖𝑥𝑡 + 𝑏𝑖𝑖 +𝑊ℎ𝑖ℎ𝑡−1 + 𝑏ℎ𝑖) +(3) +𝑓𝑡 = 𝜎(𝑊𝑖𝑓 𝑥𝑡 + 𝑏𝑖𝑓 +𝑊ℎ𝑓 ℎ𝑡−1 + 𝑏ℎ𝑓 ) +(4) +𝑔𝑡 = 𝑡𝑎𝑛ℎ(𝑊𝑖𝑔𝑥𝑡 + 𝑏𝑖𝑔 +𝑊ℎ𝑔ℎ𝑡−1 + 𝑏ℎ𝑔) +(5) +𝑜𝑡 = 𝜎(𝑊𝑖𝑜𝑥𝑡 + 𝑏𝑖𝑜 +𝑊ℎ𝑜ℎ(𝑡 − 1) + 𝑏ℎ𝑜) +(6) +𝑐𝑡 = 𝑓𝑡 ⊙ 𝑐𝑡−1 + 𝑖𝑡 ⊙ 𝑔𝑡 +(7) +ℎ𝑡 = 𝑜𝑡 ⊙ 𝑡𝑎𝑛ℎ(𝑐𝑡) +(8) +The main advantage of LSTM to other neural networks in tem- +poral feature understanding is the memory cell 𝑐𝑡, which is used +to accumulate state information in each time step. To decide what +to remember and forgot, Equation 3 and 4 calculate the input gate +and forget gate value, respectively. The input gate 𝑖𝑡 decides which +information (which is calculated by Equation 5) is saved to the +memory cell. On the other hand, the forget gate 𝑓𝑡 control which +part of the previous cell status could be forgotten. With these calcu- +lations, we determine what is the new memory cell status through +Equation 7. Additionally, how the memory cell 𝑐𝑡 propagates to the +final state or output ℎ𝑡 (through Equation 8) is controlled by the +output gate 𝑜𝑡 (calculated at Equation 6). This design allows the +LSTM to take into account previous state information and can be +self-learned through the training process. In these equations, 𝑥𝑡 is +the data at time step 𝑡, 𝑏 is the bias in each network connection, +the upper case 𝑊𝑖 and 𝑊ℎ represents the matrices of the weight of +the input data and recurrent connection, respectively, and ⊙ is the +Hadamard product. +The architecture of LSTM is shown in Figure 7. It has an input +size of N. 3 LSTM layers are stacked with 256 hidden units. They +are followed by a linear layer with an input size of 256 and an +output size of 2. A Softmax layer is added after the linear layer. The +training uses a cyclic learning rate with a 5e-4 initial learning rate +with maximum epochs of 650 with the patience of 200. For loss +function, we use cross-entropy loss. We have a dropout of 0.5 in +the LSTM layers. +The sequence length in LSTM for each sample (or, a window) +needs to be the same. However, we observe burst of packet losses +in nLoS conditions. As a result, the number of packets varies from +windows to windows (in 3 seconds). Hence, the length of the LSTM +sequence needs to be determined. We take the median of the number +of packets of the windows of the training set, which is 855 packets +and set that the sequence length of LSTM. If there are more packets, + +30 +Power Delay Profile +8 +52050 +6 +4 +5 +-17-15 +-14-10 +-23 +6 +9 +Rider distance from the car (meters)30 +525 +Power Delay Profile +15.0 +12.5 +10.0 +7.5 +10 +5.0 +5 +2.5 +0 +-19 +-17-14-12-10 +0 +2 +61113 +17 +Rider distance from the car (meters)LSTM 1  +(256) +LSTM 2  +(256) +LSTM 3 +(256) +… +𝑍� +𝑍� +FC (256) +Log softmax +log � exp �𝑍�� +∑ exp �𝑍�� +� +� +1/0 +Classification +Result +Amplitude Diff (N) +PDP (M) +Multipath (1) +FC (2) +Figure 7: LSTM architecture +then we ignore the rest. If there are fewer packets, then we perform +zero padding at the end of the sequence. In this way, we actually take +1.5 seconds of Wi-Fi packets half of the time for the classification. +Before feeding the CSI amplitude difference, power delay profile, +and multipath profile features to LSTM, we normalize them. This +is important to make sure different features with different scales +(especially the dominant multipath) do not force the network to +weigh differently. So, the features from the multipath profile and +power delay profile need to be crafted in a way that even after nor- +malization, the distinction of LoS and nLoS does not disappear. In +order to ensure that, we create just one feature using the multipath +profile by dividing the magnitude of the dominant multipath with +that of the less dominant multipath. For the power delay profile, +adding 60 input channels to LSTM may cause over-fitting. So, we +apply Principal Component Analysis of the 60 PDP features and +take the top M principal components to feed to the network. We +vary M from 3 to 5 and show the results in the Evaluation sec- +tion. After this process, the normalization retains the LoS and nLoS +distinction and reduces the number of input channels to LSTM to +reduce overfitting. +6 +DATA COLLECTION +6.1 +System Setup +To extract the Wi-Fi data in an automotive environment, we utilize +a laptop with an Intel 5300 Wi-Fi Network Interface Card (NIC) for +portability, as shown in Figure 8. We use the Linux CSI tool [12] to +collect PHY layer CSI information from received Wi-Fi packets. The +car is driven with this set up for receiving Wi-Fi packets about 10-20 +miles per hour. The three antennas are placed in the dashboard. We +mark them as A (leftmost), B (middle), and C (rightmost), when +viewed from inside of the vehicle. Although we collect data with 3 +antennas, we use only two antennas for our approach (antennas +A and C). On the rider side, the rider stands with a Pixel 2 XL +phone that serves as an Access Point (AP) at 5 GHz to which the +laptop is connected to. An Android app from the phone generates +Wi-Fi traffic by pinging the laptop as shown in Figure 9(a), that we +Table 1: Distribution of 85 rides under different conditions. +Rider left side +Rider ride side +Only Rider +7 +6 +People both sides of the rider (no car) +5 +6 +Two other people blocking signal +13 +14 +Two cars blocking signal (no other people) +10 +12 +Two cars and three other people blocking signal +6 +6 +developed and can achieve a packet transmission rate of up to 350 +packets per second. +6.2 +Collected Dataset +In order to consider realistic scenarios with LoS and nLoS condi- +tions, we collect data of 85 rides under five different conditions: (a) +only rider standing, (b) people standing on both sides of the rider, +(c) two other people blocking the signal, (d) two other parked cars +blocking the signal, and (e) two other cars and three other people +blocking the signal. We collect data when the rider is on the left +and right sides of the car in all these conditions. Table 1 shows the +number of rides under different conditions. Figure 10 shows a drone +image of the case (e), where the rider is standing, and three people +and two cars are blocking the signal. The Wi-Fi receiving unit is in +the blue car that was being driven from the left to the right side. +We split the dataset into training (60%), validation (20%), and +testing (20%). We do this per for each condition and for each side +of the rider. For example, when the rider is at the left side and two +cars blocking the signal, we have 10 such rides. We take CSI data +of 6, 2, and 2 rides for training, validation, and testing, respectively. +In that way, the test set has data of disjoint rides and under all +conditions. For each ride, we split the sequence of CSI values into +a 3 seconds window with 0.4 seconds stride length. This gives us +1032 windows for training, 286 windows for validation, and 285 +windows for testing. +6.3 +Ground Truth Collection +In order to collect the ground truth of whether the rider is at the +left or right side of the car, one can just record the timestamps of +received packets for each side of the rider. However, we would like +to collect the (x,y) location of the car when each packet was received +to have a better understanding of how the CSI changes when the +vehicle approaches the rider and leaves the rider at each side. In +order to achieve this goal, we use an off-the-shelf consumer drone +hovering above the data collection site to record the process. An +example frame from the recorded video is shown in Figure 10. Before +the data collection, we first determine landmark locations (e.g., the +rider’s location) and four positions that can form a rectangle area +with tape-measured ground truth coordinates. Next, we place solid +red-colored papers at each location and on top of the car to enable +simple color-based pixel tracking through color thresholding. In the +recorded video, we use the four locations to perform Homography +transformation so that the pixel plane and real-world plane are +parallel. This transformation creates a straightforward translation +from the pixel coordination system to the real-world coordination +system through scaling. Then we can track the vehicle in the pixel +domain and interpolate the real-world (x,y) location through the + +@ BOSCHFigure 8: In-vehicle system setup. The left picture shows the antenna viewed from outside. The middle picture shows Wi-Fi antennas placed +on the central dashboard. Antennas from the left to the right are labeled as 𝐴, 𝐵, and 𝐶. The right picture shows a laptop with an Intel 5300 +NIC and connected with antennas through cables. +(a) Android App developed to gener- +ate Wi-Fi traffic. +(b) View of how the phone is held rel- +ative to the car. +Figure 9: Data collection equipment and environment. +translation. Prior to each data collection, we also time-synchronize +the Android phone, the laptop with the Intel Wi-Fi chipset, and the +drone. The time-synchronization between the phone and drone is +achieved by capturing the phone’s time with millisecond accuracy +at the beginning of each drone’s video; thus, we can calculate the +timestamp based on the frame rate and a reference frame that has +the phone’s time clearly recorded. We also capture a screenshot +with both the phone’s time (through the laptop’s camera) and the +laptop’s time displayed with millisecond accuracy; thus, the time +difference between them can be easily calculated. We apply these +time offsets to change the timestamp recorded on the laptop and +the drone to match the time on the phone for time synchronization. +7 +EVALUATION +In this section, we estimate the accuracy of CarFi and compare it +with state-of-the-art methods. We investigate the effect of antenna +spacing, subcarrier selection, and window size on the performance +of the solution. Also, we estimate its execution time, and range in +both LoS and nLoS conditions. +7.1 +Accuracy +To the best of our knowledge, there is no state-of-the-art Wi-Fi- +based rider side determination technique. So, we implement a few +Figure 10: Drone image of three people and two vehicles in between +the Wi-Fi transmitter and receiver. +baseline methods to compare with our approach in terms of accu- +racy. +Baseline 1: CSI phase difference based approach: Although +our approach does not require phase calibration, in order to inves- +tigate and compare with a phase difference based approach, we +perform phase calibration of the antenna chains of the Intel 5300 +chipset attached with the laptop using a method similar to [50]. We +use another laptop with Intel 5300 chipset to transmit Wi-Fi packets +through an RF splitter, where all CarFi three receiver antennas are +connected to the RF splitter’s output as shown in Figure 11. These +three antennas should receive the Wi-Fi signal at the same time. +However, due to the slight path distance difference within the RF +splitter, we switch the receiver antennas’ connecting locations and +record the phase information in each connection combination to +eliminate the difference introduced by the RF splitter. By removing +the offset we measured, we correct the antenna phase offset in our +collected data. The system also introduces Sampling Time Offset +(STO) and Sampling Frequency Offset (SFO) as the sampling clocks +and frequencies are unsynchronized between the receiver and the +transmitter. We follow SignFi [25] to remove STO and SFO through +multiple linear regression. +We use only antennas 𝐴 and𝐶, and estimate the phase difference +by subtracting unwrapped phase𝐴 from unwrapped phase𝐶 of each +window. Ideally, the phase difference should be positive (negative) +when the rider is on the left (right) side. But for 30 different sub- +carriers, the patterns vary significantly. As an example, we show the +phase difference when the rider is at the right side in LoS condition +and when he was blocked by two cars in Figure 12. + +Figure 11: Antenna chain phase calibration of the Intel 5300 chipset. +(a) Unwrapped phase difference of antennas (𝐶 - 𝐴) in LoS condition. +(b) Unwrapped phase difference of antennas (𝐶 - 𝐴) in nLoS condition. +Figure 12: Unwrapped phase difference in LoS and nLoS conditions +when the rider is in the right side. +Figure 13: Positive and negative votes of unwrapped phase differ- +ence between antennas when the rider is in the both sides. +Figure 14: Estimating effective phase difference between antennas +(𝐶 - 𝐴) when the rider is at the right side in LoS. +Since the unwrapped phase difference change over time, we +consider four different ways to compute the features to capture +phase difference between antennas: +(a) We average all phase differences of all sub-carriers of all +packets within a window. The intuition is that mean of phase +difference should be different for different sides. +(b) Similar to (a), but instead of all the sub-carriers, we use just +the first sub-carrier. +(c) We divide the window into a few sub-windows. The reason +for sub-windowing is to reduce the propagation error of +phase unwrapping. Then, we average all phase differences +of all sub-carriers in each sub-window. We remove 20% sub- +windows with large variance. Then, we compute a positive +or negative vote for each sub-window based on the sign of +its phase difference. We count the numbers of positive and +negative votes, and use them as features. We plot the num- +ber of positive and negative votes for all the 1032 training +windows and plot them in Figure 13. The green line shows +when the positive and negative votes are the same. It shows +that when the rider is on the left (right) side, there are more +positive (negative) votes (as they should be). We calculate +such votes for all the sub-carriers. +(d) After sub-windowing, we compute an effective phase differ- +ence for each subcarrier. The intuition is that phase differ- +ence should be stable for each subcarrier because the central +frequency is the same. We choose an effective phase differ- +ence that covers most phase differences within two radians +and has the smallest mean error. An example of such an +effective phase difference is shown in Figure 14 when the +rider is on the right side and in LoS condition. It shows that +even though the phase difference fluctuates, the effective +phase difference is negative (as it should be since the rider +is on the right side). We estimate like this for 30 sub-carriers +and use all 30 effective phase differences as features to the +classifiers. +We feed these features to kNN, DT, and SVM classifiers and show +the results of rider side classification in Table 2. For kNN, we vary +the value of k from 3 to 15 and report the accuracy with the best +k. We see the highest accuracy we get from the phase difference +based approach is only 56%. +Baseline 2: RSS difference based approach: When we collect +data, we also collect RSS (Received Signal Strength) values from +each antenna. We feed the average RSS difference of antennas (𝐶 - +𝐴) of each window to different classifiers, including KNN, Decision +tree, and SVM, to classify the rider side. The results are shown in +Table 2. The results show that the highest accuracy is 85.6% that +came from both K-NN and SVM. It provides higher accuracy than +the CSI phase difference based approach. +Baseline 3: CSI amplitude difference based approach: +Since the CSI amplitude difference changes over time, we con- +sider different ways to compute features to capture amplitude dif- +ference of antennas (𝐶 - 𝐴): +(a) We average all CSI amplitude differences of all sub-carriers +of all packets within a window. +(b) Similar to (a), but we use only the first sub-carrier. +(c) Similar to (a), but we also add average RSS difference. +(d) Similar to (b), but we also add average RSS difference. +We feed the features to kNN, DT, and SVM classifiers. The results +are shown in Table 2. Its shows the highest accuracy is 89.5%, when + +CSI phase difference (rad) +100 +Subcarrier 1 +Subcarrier 16 +Subcarrier 2 +Subcarrier 17 +50 +Subcarrier 3 +Subcarrier 18 +Subcarrier 4 +Subcarrier 19 +Subcarrier 5 +Subcarrier 20 +0 +Subcarrier 6 +Subcarrier 21 +Subcarrier 7 +Subcarrier 22 +50 +Subcarrier 8 +Subcarrier 23 +Subcarrier 9 +Subcarrier 24 +Subcarrier 10 +Subcarrier 25 +100 +Subcarrier 11 +Subcarrier 26 +Subcarrier 12 +Subcarrier 27 +0 +200 +400 +600 +800 +1000 +Subcarrier 13 +Subcarrier 28 +numberof packets +Subcarrier 14 +Subcarrier 29 +Subcarrier 15 +Subcarrier 30Iphase difference (rad) +80 +Subcarrier: +Subcarrier 16 +60 +Subcarrier 2 +Subcarrier 17 +40 +Subcarrier +Subcarrier 18 +Subcarrier 4 +Subcarrier 19 +20 +Subcarrier 5 +Subcarrier 20 +Subcarrier 6 +Subcarrier 21 +Subcarrier 7 +Subcarrier 22 +20 +Subcarrier 8 +Subcarrier 23 +40 +Subcarrier +Subcarrier 24 +Subcarrier 10 +Subcarrier 25 +60 +Subcarrier 11 +Subcarrier 26 +80 +Subcarrier 12 +Subcarrier 27 +0 +100 +200 +300 +400 +500 +600 +700 +800 +Subcarrier 13 +Subcarrier 28 +numberofpackets +Subcarrier 14 +Subcarrier 29 +Subcarrier 15 +Subcarrier 30f negative votes +250 +Rider in the left side +Rider in the right side +200 +150 +Numberof +100 +50 +0 +50 +100 +150 +200 +250 +NumberofpositivevotesPhase difference of subcarrier 1 +(rad) +10 +Estimatedeffectivephasedifference +difference +5 +Phase +10 +15 +0 +200 +400 +600 +800 +1000 +NumberofpacketsTable 2: Results of left vs. right classification of baseline methods. +Baseline +KNN (%) +DT(%) +SVM(%) +1(a) +50.2 (k=5) +46.0 +48.4 +1(b) +54.4 (k=11) +50.2 +44.2 +1(c) +52.6 (k=3) +52.6 +51.6 +1(d) +52.6 (k=3) +56.0 +49.5 +2 +85.6 (k=10) +76.1 +85.6 +3(a) +84.2 (k=5) +81.1 +84.6 +3(b) +88.1 (k=3) +83.9 +87.7 +3(c) +83.5 (k=7) +81.4 +85.3 +3(d) +87.3 (k=8) +82.5 +89.5 +combining the average CSI amplitude difference and average RSS +difference. +We also implement our LSTM based network and change net- +work parameters, including the size of hidden dimensions and +number of layers to see how that affects performance. The results +are shown in Table 3. It shows that when we use our variance +based subcarrier selection, the accuracy is higher than when all sub- +carriers are used, or only the first subcarrier is used. We see that we +get 95.44% accuracy when we combine variance based subcarrier +selection, power delay profile, and multipath profile. This highest +accuracy came from when we select 14 subcarriers with VbSS, ob- +tain 3 PDP features and 1 multipath profile feature. If we feed the +exact same features to kNN, DT, and SVM, we get 68.4%, 69.5% , +84.2% accuracy, respectively. Hence, our LSTM based architecture +increases accuracy by 11.24% from exactly the same input. +7.2 +Sensitivity Analysis +In this section, we analyze the effect of antenna spacing, subcarrier +selection, and window size on CarFi performance. +Effect of antenna spacing: In our analysis, the default antenna +spacing was 5.2 cm, which produced 95.44% accuracy. Since we +collected data with three antennas, we can use antenna 𝐴 and 𝐵 to +see how the performance looks like when the antenna spacing is +2.6 cm. We keep the best performing network’s parameter the same +and run the experiment with 2.6 cm spacing and find the accuracy +is only 55.79%. Thus, increasing antenna spacing helps improving +accuracy. +Effect of window size: In our analysis, the default window size +is 3 seconds. We keep the best performing network’s parameter the +same and run the experiment by varying window sizes to 0.5, 1, +1.5, 2, 2.5, and 3 seconds find the accuracy is 62.95%, 79.36%, 80%, +85.17%, 89.03%, 95.44%, respectively as shown in Figure 15. We see +that longer windows provide higher accuracy. +Effect of number of sub-carriers: We keep the best perform- +ing network’s parameter the same and run the experiment with +changing the number of sub-carriers from 1 to 16 and show the +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Window Size (seconds) +50 +60 +70 +80 +90 +100 +Accuracy (%) +Figure 15: Effects of window size on accuracy. +0 +2 +4 +6 +8 +10 +12 +14 +16 +Number of subcarriers +86 +88 +90 +92 +94 +96 +Accuracy (%) +Figure 16: Effects of number of sub-carriers on accuracy. +impact of the number of sub-carriers through our VbSS method on +accuracy in Figure 16. When we choose only one subcarrier, we +choose subcarrier 1, as it provides 89.47% accuracy. We achieve the +highest accuracy (95.44%) when the number of sub-carriers is 14. +7.3 +Execution Time +We train our LSTM using Nvidia GeForce GTX 1080 Ti GPU. It +takes about two hours to train the network. However, the infer- +ence is rapid. We estimate how long it takes to perform inference +in a powerful GPU like NVidia GeForce GTX 1080 Ti as well as +an embedded GPU like Nvidia Jetson Nano. It takes only 101.77 +and 850.37 milliseconds to execute the inference process in 1080 +Ti and Jetson Nano, respectively. Hence, the solution can be run +on embedded GPUs in real-time. Also, there are several ways to +optimize (e.g., recompiling with TensorRT can significantly reduce +inference time on Jetson devices) and prune the model to compress +the network, which will reduce inference time [53] [46]. We leave +this to future work. +7.4 +Range Analysis +In this section, we estimate how far CarFi can operate in both LoS +and nLoS conditions. We collect additional data for this evaluation. +We have a person standing at different distances ranging from +10 meters to 120 meters in front of the car in both LoS and nLoS +conditions. We transmit 10,003 packets from each location. To create +a nLoS condition, we have a person standing between the phone +and Wi-Fi receiving unit placed in the car dashboard. The Packet +Delivery Ratio (PDR) at different distances from the car is shown +in Figure 17. We see that even at a range of 120 meters, the PDR +is 99.30% in LoS condition. However, in nLoS situation, the PDR +drops sharply to 41.65% in 70 meters. At 120 meters, the PDR is + +Table 3: Results of left vs. right classification when LSTM with different features are used. +Description +Input Dim +Hidden Dim +Number of layers +Optimizer +Accuracy +Variance-based Subcarrier Selection (VbSS) +12 +256 +3 +RMSProp +93.33% +Variance-based Subcarrier Selection (VbSS) +12 +256 +3 +Adam +91.57% +Variance-based Subcarrier Selection (VbSS) +12 +128 +3 +RMSProp +92.28% +Variance-based Subcarrier Selection (VbSS) +12 +256 +4 +RMSProp +89.82% +Variance-based Subcarrier Selection (VbSS) +12 +128 +4 +RMSProp +92.63% +First Subcarrier only +1 +256 +3 +RMSProp +89.47% +All sub-carriers +30 +256 +3 +RMSProp +89.47% +VbSS + Multipath Profile (MP) +12+1 +256 +3 +RMSProp +93.33% +VbSS + Power Delay Profile (PDP) +12+5 +256 +3 +RMSProp +93.33% +VbSS + Power Delay Profile (PDP) +12+3 +256 +3 +RMSProp +93.68% +VbSS + PDP + MP +12+3+1 +256 +3 +RMSProp +95.08% +VbSS + PDP + MP +14+3+1 +256 +3 +RMSProp +95.44% +10 +20 +30 +40 +50 +60 +70 +80 +90 100 110 120 +Rider distance from the car (meters) +0 +20 +40 +60 +80 +100 +Packet Delivery Ratio (%) +LoS +nLoS +Figure 17: Packet delivery ratio at different distances from the car. +only 0.75%. So, we see that in LoS conditions, CarFi will operate +beyond 120 meters range. +8 +DISCUSSION +8.1 +Generalizability +Although the data was collected from one large parking lot, we put +an effort to introduce variation in the rides by asking the volunteers +to stand differently to block the signal, move while blocking the +signal, drive at different speeds, and vary the speed in different +rides. As a result, there is a significant variation in the dataset, and +we expect the model to generalize to some extent. One particular +reason we were not able to collect data from a busy street is that +the Wi-Fi of the laptop needed to stay connected to the phone +for data collection with CSI Tool [13], which is very difficult to +obtain in busy streets as the car can easily go out of the Wi-Fi +range. Currently, we are switching to Nexmon framework [37] +for collecting CSI data, where the phone will inject packets at a +particular Wi-Fi channel. This will allow us to perform a large-scale +data collection from busy streets for testing the generalizability of +the solution. We leave it to future work. +8.2 +Limitations +One potential limitation of our approach is that if someone else +books the ride on behalf of the rider and the rider has a different +phone, then the proposed solution may not work. If the rider does +not co-operate (e.g., by disabling Wi-Fi), then it will not work. Also, +it will need support from the rider-hailing service providers (e.g., +Uber, Lyft) to enable this service into their apps. However, they can +incorporate such a service using their apps and dashboard products. +Also, our approach assumes that the rider is waiting for the car on +a side of a street while the car is moving towards or away from the +rider. If the rider books the trip from inside, e.g., from a restaurant, +and waits inside for the car to arrive, then we will not be able to +leverage the motion related features as effectively, and it may not +be able to assist the driver with the rider side. However, when the +rider comes out of the restaurant, we can ask him to walk along his +side of the street to help us capture some features to determine his +side. Due to the time sensitive nature of the application, assuming +it takes maximum 3.85 seconds (0.85 seconds for execution and +maximum 3 seconds of window, although 50% time we only need +1.5 seconds to collect enough packets and perform classification) to +run CarFi on a Jetson Nano and Wi-Fi range of 70 meters in nLoS, +CarFi can only determine the rider side when the rider is in front +of the car if the vehicle approaches towards the rider at 40.67 miles +per hour or less. However, it can work up to 81.34 miles per hour +vehicular speed if we are not required to determine rider side while +the rider is in front of the car. +8.3 +Lessons Learned and Future Work +Our exploratory study shows that the phase difference between +antennas in an automotive environment is very noisy and does not +provide as high accuracy as amplitude difference-based methods +for determining rider side. We also find that increasing the antenna +distance even by 2.6 cm helps improving accuracy for the amplitude- +based approach significantly. The maximum length of an object +that can be placed in a dashboard is 5.5 inches by the laws of + +California, USA. We plan to increase antenna spacing and see if +we can achieve higher accuracy at smaller window sizes within +this constraint. We leave this to future work. Also, we estimate +the rider side independently from each window. In the future, we +can take into account all the packets of the past windows from the +same ride, assuming that the rider did not cross the street while +the car is approaching and have a bigger window for rider side +determination. Also, we plan to collect data from busy streets and +evaluate the performance in challenging scenarios. +9 +RELATED WORK +9.1 +Human Detection and Identification +There has been a significant amount of work in detecting the pres- +ence of humans and identifying them. For the ride-hailing appli- +cation, the vehicle needs to detect the presence of the rider and +identify him/her before localizing him/her. Humans can be detected +using cameras [48, 54], depth sensors [10, 11, 20, 21, 26, 28–30], IR- +array sensors [27], mmWave radars [5, 52], Wi-Fi [4, 7–9], and +using other sensors for various purposes [14, 33, 34]. +Vision-based systems such as [39, 43, 44] can be used to identify +the person but requires prior knowledge such as the facial features. +These systems are privacy invasive (e.g., facial recognition systems +are being banned in multiple cities for privacy concerns) and gener- +ally do not work if the subject is at a distance and can be occluded. +Systems such as ID-Match [19] uses RFID tags and a 3D depth cam- +era, FORK [28] uses a depth sensor, and [23] uses RFID and BLE +to identify individuals. These works require additional sensors or +devices, which are not practical in the ride-hailing scenario, difficult +to scale, and potentially costly. Similar to EyeFi [7] and RFCam [4], +we utilize the phone as an identifier as it is already being utilized +in the ride-hailing application. +9.2 +Localization +Numerous works have focused on localization using Wi-Fi. [18] +exploits Wi-Fi subcarrier CSI values to create virtual antennas to +obtain higher resolution Angle of Arrival (AoA) estimation, then +using multiple Wi-Fi receivers to triangulate the transmitter for +localization. The usage of multiple Wi-Fi receivers is expensive, and +AoA estimation is unstable in our automotive environment. [49] +proposes a higher resolution of power delay profile by utilizing CSI +splicing. This method requires a special Wi-Fi configuration, which +is impractical for our application. However, it does inspire us to +use power delay profile as one of our features. Other works such +as [38, 41, 42, 45] have examined using time-of-flight, frequency, +and multipath to estimate AoA and triangulate the locations of the +transmitter. These methods are more suited for indoor settings and +more prone to environmental changes. +Using simple features such as RSSI has been explored in previ- +ous works to perform vehicle localization [6]. Similarly, [31] uses +a fingerprinting method to localize vehicles in a car park. While +fingerprinting is simple and easy to perform, they are prone to en- +vironmental changes and lacks generalization ability as it requires +prior knowledge for each place. [51] locates a bus by scanning Wi-Fi +APs surrounding the bus route and predicts the time for the bus’s +arrival. However, it still lacks the generalization ability and can not +provide fine-grained location information regarding the rider side. +One work that is closely related to our work is [15], which +uses Wi-Fi Fine Time Measurement (FTM) to measure the distance +(through ToF) and achieve high precision localization. However, is- +sues such as the time needed to perform such measurement (which +can take several seconds), the requirement of the phone to con- +nect, and the need to place antennas on both sides of the vehicle’s +roof prevent it from being conveniently deployed. Similarly, [32] +utilizes both CSI and FTM to achieve higher accuracy with added +AoA (Angle of Arrival) and AoD (Angle of Departure) measure- +ments. However, this method still depends on FTM that most of the +smartphones do not support. +9.3 +Wi-Fi Sensing +While some work in Wi-Fi sensing does not directly apply to local- +ization problems, they can provide meaningful features that may +be utilized. For example, [47] use Wi-Fi CSI phase change dynamics +with Fresnel zone model to determine the human subject’s walk- +ing direction. The phase change dynamics contain environmental +changes around the transmitter and receiver. In [16], the author +demonstrated how the Wi-Fi signal contains detailed information +that can be used to reconstruct human pose, and a neural network +can be used to extract deeper features. In these works, the transmit- +ters and receivers are stationary. However, in our case, the receiver +is moving. +10 +CONCLUSION +In this paper, we investigate the feasibility of using Wi-Fi based +street side determination of riders from a car to assist drivers to +locate their riders. This method can potentially enable a smoother +pick-up experience. Our approach uses a two-antenna Wi-Fi chipset +for this purpose. After extracting CSI values, it computes relevant +features by leveraging the motion of the car and utilizes a data- +driven technique to determine the rider side on an embedded GPU +in real-time. By performing extensive evaluation in the real-world +in both LoS and nLoS conditions, we see CarFi achieves 95.44% +accuracy for estimating the rider side. 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IEEE, 2006. + diff --git a/edAzT4oBgHgl3EQfoP0g/content/tmp_files/load_file.txt b/edAzT4oBgHgl3EQfoP0g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..55cf4853beda2a84abc3d95646094ddc2b76bbcc --- /dev/null +++ b/edAzT4oBgHgl3EQfoP0g/content/tmp_files/load_file.txt @@ -0,0 +1,1067 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf,len=1066 +page_content='CarFi: Rider Localization Using Wi-Fi CSI Sirajum Munir∗† Bosch Research and Technology Center Pittsburgh, PA, USA sirajum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='munir@us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='bosch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='com Hongkai Chen∗ Shan Lin Stony Brook University Stony Brook, NY, USA {hongkai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='chen,Shan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Lin}@stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='edu Shiwei Fang∗ Mahathir Monjur Shahriar Nirjon UNC Chapel Hill Chapel Hill, NC, USA {shiwei,mahathir,nirjon}@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='edu ABSTRACT With the rise of hailing services, people are increasingly relying on shared mobility (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', Uber, Lyft) drivers to pick up for transporta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, such drivers and riders have difficulties finding each other in urban areas as GPS signals get blocked by skyscrapers, in crowded environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', in stadiums, airports, and bars), at night, and in bad weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' It wastes their time, creates a bad user ex- perience, and causes more CO2 emissions due to idle driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In this work, we explore the potential of Wi-Fi to help drivers to determine the street side of the riders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Our proposed system is called CarFi that uses Wi-Fi CSI from two antennas placed inside a moving vehicle and a data-driven technique to determine the street side of the rider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' By collecting real-world data in realistic and challenging settings by blocking the signal with other people and other parked cars, we see that CarFi is 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='44% accurate in rider-side determination in both line of sight (LoS) and non-line of sight (nLoS) conditions, and can be run on an embedded GPU in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 1 INTRODUCTION As people rely on ride-hailing services, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', Uber and Lyft, it be- comes increasingly important for drivers and riders to find each other without a hitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Currently, drivers and riders use smartphones, which rely on GPS or cellular signals, to locate each other while far apart, and require them to recognize each other while nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, in urban cities and areas like downtown, where there are numerous skyscrapers, GPS signals often do not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In addition, there are places, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', in airports, where the drivers need to come indoors (such as parking garages) to pick up riders where the build- ing structure blocks GPS signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Also, it is challenging to locate the actual rider among many people in crowded environments like stadiums, airports, theatres, and bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Moreover, the situation can worsen due to lack of visibility, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', at night and during bad weather (such as rain, storm, and snow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' This issue wastes the time of the riders and drivers, causes more CO2 emissions due to idle driving, causes frustration, and creates a bad user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' A recent Uber study shows that every user agreed that they do not like to negotiate the pickup point, and 11 out of 16 users find it hard to give directions to the driver when the user is at a new place [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Based on our discussion with a few drivers and riders, we find that determining the street side of the rider is very crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' This is because, if the car is on the other side of the street, the rider sometimes must cross the street, which can be unsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Also, the drivers do not want to make a U-turn and realize that they were on ∗Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' the right side in the first place, which leads to a double U-turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' So, we focus on determining the street side of the riders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Several solutions have been proposed to improve the rider pick- up experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' For example, the vehicle can use a camera and facial recognition [35] to identify the rider and subsequently compute the location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, facial recognition requires the rider to upload his or her photo, which can be privacy-invasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Moreover, for facial recognition to work, the rider needs to be within the camera’s field of view and occupy enough pixels to be successfully recognized and have good lighting conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' One can also ask the user to scan the surroundings with his or her phone, and then a server can perform 3D reconstruction [17] and matching [22] to the previously established real-world model to compute the exact location of the rider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, this is a computation-intensive approach, and this method also requires the world to be digitized and constructed to allow such matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' As commercial products, Uber and Lyft have multicolored LED-based lights for riders to recognize their cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, such a solution does not work in broad daylight, and it is a rider-oriented solution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', the rider has to find the car, and the driver does not have much information about the location/side of the rider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Our proposed solution overcomes these limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In this paper, we perform an exploratory study to understand the feasibility of using Wi-Fi to address this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We propose to utilize smartphones – which the riders will mostly like to possess for the ride-hailing booking – and Wi-Fi-enabled dashcams – in- creasingly crucial for safety and legal purposes – to determine the street side of the rider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We call our proposed system CarFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' CarFi neither requires the rider to upload any photos of him or her nor a photo of the surrounding area, which protects the rider’s privacy, reduces the computation load, and does not depend on lighting conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' To this end, CarFi uses Wi-Fi communications between the rider’s smartphone and the dashcam onboard the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The usage of the dashcam is for the purpose of standalone devices that can be installed on any vehicle, but the proposed system does not exclude vehicles that have Wi-Fi already installed to localize the rider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' CarFi uses a two-antenna Wi-Fi chipset embedded in the dash- cam to receive the Wi-Fi packets sent by the smartphone held by the rider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' This system does not require any modification to the vehicle and the smartphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The Wi-Fi packets can be generated by the ride-hailing app, which can share the phone’s MAC address (or, a randomized MAC address) through the cloud/server to the vehicle (or, driver’s app).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Thus, the vehicle can listen to the packets generated from the target phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The system on the vehicle extracts the Channel State Information (CSI) data from the Wi-Fi chipset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='01592v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='NI] 21 Dec 2022 After some preprocessing, it performs sub-carrier selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Then, it extracts relevant features (amplitude difference between antennas, multipath profile, power delay profile) for rider-side determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Then, the contextual and motion-related features are encoded into a data-driven model (LSTM) to classify whether the rider is on the right or the left side of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Our approach only uses CSI amplitude and does not use CSI phase information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Thus, it avoids effort for phase calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' This work has the following contributions: First, we perform a comprehensive exploratory analysis to understand the potential of using Wi-Fi CSI in an automotive environment for shared mobility applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Our empiri- cal study involves determining the set of features that can effectively work in an automotive environment in both line of sight (LoS) and non-line of sight (nLoS) conditions when a vehicle is being driven and encoding the features into the design and implementation of a data-driven model (LSTM) for estimating the side of the rider using only two anten- nas and CSI amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Our CarFi system does not require privacy-invasive personal information from the rider such as a photo, avoids heavy computation on the server, and works in the dark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Second, we set up an infrastructure to collect Wi-Fi CSI from a moving vehicle with a drone-based system for annotating the ground truth location of the vehicle when each packet is received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We collect a dataset of 85 rides with over 568,000 Wi-Fi packets in a realistic and challenging environment, considering both LoS and nLoS, where other people and other parked vehicles block Wi-Fi signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' To the best of our knowledge, this is the first dataset to investigate how Wi-Fi CSI changes over time when the receiver is placed inside a moving car for shared mobility applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Third, based on evaluation using data collected from the real-world, our results show that CarFi is 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='44% accurate in classifying the rider side in both LoS and nLoS conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We also implement several baseline solutions using phase difference and other features and show the superiority of our solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We also evaluate the execution time of our approach in both powerful and embedded GPUs and show that our solution can be run on an embedded GPU in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 2 CARFI OVERVIEW An overview of the CarFi system is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' When a rider wants to travel to a specific location, s/he uses the ride-hailing phone app on the phone to book a trip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The cloud server of the service providers processes the request and finds a driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The loca- tions of the vehicle and the rider are determined by their respective location providers, such as GPS on the phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Once the trip is con- firmed, the driver heads toward the rider’s location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' As the driver arrives within a certain distance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 miles from the rider based on the location data, the rider’s phone will transmit Wi-Fi packets at a higher transmission rate as the ride-hailing app controls it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In the meantime, the phone’s MAC address is shared with the dashcam via the servers in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' A randomized temporary MAC address can be used to preserve the privacy of the rider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' As the vehicle is also within this certain range, the dashcam starts listening for Wi‐Fi Packets Filtering Feature Extraction Pre‐ processing Wi‐Fi Transmission CSI Extraction Rider’s Phone App MAC Address LSTM Left vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Right classification Driver’s Phone App Visualization of Rider’s side Figure 1: CarFi system overview Wi-Fi packets containing the phone’s MAC address and filters out other packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' When CarFi system receives the Wi-Fi packets with matched MAC address, it extracts the CSI information, performs some pre-processing, and calculates relevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Then it feeds the features to an LSTM, which estimates the street side of the rider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Then, this information is passed to the driver’s smartphone app from the dashcam for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The data exchange between the phone and the dashcam can be achieved via either Bluetooth or cellular connection (if the dashcam has it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 3 CHALLENGES In this section, we discuss the challenges that CarFi system faces for rider side localization in an automotive environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='1 Automotive Environment When moving Wi-Fi devices from indoor locations to automotive environments, the characteristics of the environment and its effects on the signals change dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' One of the biggest issues in an automotive environment is the metal structure of the vehicle body, which can be similar to a Faraday cage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Although the signal of normal radio frequency communication systems has a higher frequency than what the window can block due to its large size, the vehicle’s metal surface can still block and redistribute the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Unfortunately, there has not been much work to understand how Wi-Fi CSI looks like inside of a vehicle when the vehicle is being driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' With such a complex RF environment, the current state-of-the- art method, such as SpotFi [18], can not accurately estimate the Angle of Arrival (AoA) of the Wi-Fi signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' An example of such an AoA estimation is shown in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The X-axis represents the distance of the rider from the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The car is coming from the left side of the X-axis, meets the rider in the center, and then leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The three antenna arrays are placed at the center of the dashboard of the car, and the AoA should be 0 to -90°(0 to 90°) when the rider is at the right (left) side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We consider two cases: the rider is standing without anyone blocking the signal (Figure 2(a)), and two other cars and three other people blocking the signal (Figure 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The rider was on the right side in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We see that in LoS cases, the AoA is relatively stable as the Wi-Fi signal penetrates through the front windshield, but when the car leaves the rider, there is a lot of fluctuation of the AoA as the backside of the car blocks the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We observe that when other people and cars block the rider, the D20 15 10 5 0 5 10 15 20 Rider distance from the car (m) 90 75 60 45 30 15 0 15 30 45 60 75 90 AoA (degree) AoA (a) AoA estimation of when the rider is in LoS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 20 15 10 5 0 5 10 15 20 Rider distance from the car (m) 90 75 60 45 30 15 0 15 30 45 60 75 90 AoA (degree) AoA (b) AoA estimation of when the rider is in nLoS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 2: SpotFi AoA in LoS and in nLoS conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The rider is in the right side of the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The expected AoA is 0 to -90°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' AoA is unreliable even when the rider is in front of the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Since AoA estimation also requires three antennas and phase calibration, we do not use AoA in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2 Speed and Time We do not expect that the vehicle will approach the rider at highway speed when they are nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Instead, we assume that the vehicle will be traveling at a lower speed to be able to stop quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Therefore, we assume 10 to 20 miles per hour vehicle speed, which translates to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='47 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='94 meters per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We also consider the transmission range of the Wi-Fi signal to be around 70 to 120 meters in the outdoor environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' If the rider is 70 meters in front of the car, the driver has about 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='83 to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='66 seconds to stop the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Given the human response time is about 1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 seconds, we determine that the if CarFi system takes 3 seconds, it will provide adequate time for the driver to respond and stop safely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Smartphones can transmit several hundreds of Wi-Fi packets in a second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, there could be a burst of packet loss due to non-line of sight (nLoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In addition, the more time we take to make a decision, the higher accuracy we can offer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Thus, a small window size with a variable number of received packets poses a difficult challenge for rider side determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='3 Cost In order to make the solution practical, we need to use inexpensive antennas and a lightweight computing platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' A simpler solu- tion might use two directional antennas to classify left vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, we need directional antennas with 180-degree horizontal beamwidth, which is expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' For example, [2] costs $225 per antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Cheaper ones have a smaller beamwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' For example, [3] costs $35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='94 per antenna, but has only 66 degrees horizontal beam patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Also, such directional antennas are bulky and could obstruct the field of view of the driver more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Adding more antennas also helps in improving the accuracy but also increases the cost of the Wi-Fi chipset and antenna chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Moreover, the solution needs to be lightweight to be able to run on an embedded GPU or acceler- ators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Although, such an accelerator would increase hardware cost, a dashcam with such capability could provide additional benefits to the drivers by offering additional services e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', detecting accidents, violence/aggression in the car and providing necessary support by performing audio-visual analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 4 APPROACH In this section, we describe the CarFi approach in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='1 Pre-processing When the receiving unit starts to receive Wi-Fi packets, CarFi timestamps each packet and keeps all the packets within a window size of 3 seconds for processing together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Then, it uses a stride length of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='4 seconds to create the next window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2 Feature selection In this section, we discuss the set of features that we use for left vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' right classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='1 Amplitude difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We use Channel State Information (CSI) from only two antennas for the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We assume the distance between them is 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In our exploratory analysis, we have 𝑑 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' CSI contains how the RF signal propagates through the environment as they are being affected during transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The CSI data collected at the receiver side contains those affected and encoded in the complex form with amplitude and phase infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Each CSI data point is also the Channel Frequency Response (CFR): 𝐻 (𝑓 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='𝑡) = 𝑁 ∑︁ 𝑛 𝑎𝑛(𝑡)𝑒−𝑗2𝜋 𝑓 𝜏𝑛 (𝑡) (1) Where 𝑎𝑖 (𝑡) is the amplitude attenuation factor, 𝜏𝑖 (𝑡) is the prop- agation delay, and 𝑓 is the carrier frequency [40] [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 3 shows how CSI amplitude difference between antenna 𝐶 and antenna 𝐴 looks like for a portion of a ride for 30 sub-carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The rider was on the right side of the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The X-axis shows the distance of the car with respect to the rider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The car is approaching from the left side of the X-axis, meets the rider at the middle of the X-axis, and then passes the rider after that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' When we plot amplitude difference, we plot the CSI amplitude of the antenna 𝐶 - antenna 𝐴, where antenna 𝐴, 𝐵, and 𝐶 are placed from left to right parallel to the dashboard (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' So, a positive value is a good indicator that the rider is on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We see that the amplitude difference values fluctuate over time, and they also vary for different sub- carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' While Figure3(a) shows a LoS condition, Figure 3(b) shows a nLoS condition where the three other people and two other cars were placed between the rider and the Wi-Fi receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We see a burst of packet loss there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' As the CSI amplitude varies by subcarriers, instead of relying on all the sub-carriers, we determine the relevant sub-carriers for us that are less prone to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2 Sub-carrier selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Instead of relying on all the sub-carriers, we select sub-carriers that are more resilient to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' First, we (a) Amplitude difference of antennas (𝐶 - 𝐴) when the rider is in LoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' (b) Amplitude difference of antennas (𝐶 - 𝐴) when the rider is in nLoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 3: CSI amplitude difference in LoS and nLoS conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 4: CSI Amplitude difference between antennas (𝐶 - 𝐴) of 12 selected sub-carriers using Variance-based Subcarrier Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' compute the covariance of CSI amplitude of all the subcarriers of antenna C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' High covariance between these subcarriers shows they receive effective signal and not the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' For each subcarrier in antenna C, we select the corresponding subcarrier of antenna A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' These subcarriers have similar path properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', multipath effect, attenuation) and receive correlated CSI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We vary the number of selected subcarriers from 1 to 30 and choose the number of subcarriers that provide the highest accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Please note that sub-carriers are selected per window of packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' So, different win- dows may have different sets of sub-carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We call this approach Variance-based Sub-carrier Selection (VbSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' When choosing 𝑁 subcarriers, we choose 𝑁 − 1 sub-carriers using VbSS and add the first subcarrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 4 shows the amplitude difference of the 12 selected subcarriers of a window when the rider is on the right side of the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='3 Multipath Profile Since Wi-Fi CSI data contains multipath attenuation caused by the environment, the multipath profile extracted from the CSI data can be very useful in location estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' It can effectively pro- vide whether the rider is in LoS or nLoS conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' To extract the multipath profile of the CSI data, we explore how MUSIC [36] and SpotFi [18] algorithms extract signals and estimate their Angle of Arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Inspired by this, we first isolate multiple possible signals by performing Eigen decomposition of matrix 𝑋𝑋𝐻, where 𝑋 is the CSI measurement, and 𝑋𝐻 is the conjugate transpose of 𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The eigenvectors and eigenvalues can be used as features as they are affected by the environment and the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We take the top two dominant multipaths and plot them in Figure 5 with both LoS and nLoS conditions, and the rider was on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The X-axis in (a) First and second multipaths of when the rider is LoS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' (b) First and second multipaths of when the rider is nLoS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Three other people and two other cars are blocking the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 5: Multipath profile in LoS and nLoS conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' both figures shows the distance from the car as the car is approach- ing the rider from the left side of the axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 5(a) represents the case when the rider is in a LoS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We see that as the car approaches, there is a significant difference between the first and the second multipath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, as the car leaves the rider, the backside of the car blocks the signal and causes nLoS conditions, and hence the difference between the top two multipaths decreases significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 5(b) represents the case when the rider is in a nLoS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' There were three other people and two other cars blocking the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' As a result, the first and second dominant multipath is closer from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, when the car passes the rider, it gets a line of sight for a brief moment, and hence the difference between the first and the second multipath becomes more prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We take the top two Eigenvalues representing the top two dominant multipaths computed using CSI values of two antennas as features for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='4 Power Delay Profile Power Delay Profile (PDP) describes the power level associated with each multipath along with the propagation delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, due to the limited bandwidth of the Wi-Fi channels, the path length resolu- tion is not very precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' For our 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='11ac 40 MHz channel, the path length resolution is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' But it can be helpful for coarse-grained mobility tracking over time and provide contextual information regarding LoS and nLoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' When the Wi-Fi chipset measures the channel frequency re- sponse as written in Equation 1, instead of measuring continuously, it samples the response at discrete frequency points 𝑓 = 𝑓0 + 𝑘Δ𝑓 , where k is the sub-carrier index and Δ𝑓 = 312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5𝑘𝐻𝑧 [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Since Equation 1 is in the frequency domain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' by applying Inverse Fourier Transform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' we can get the response in the time domain which is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='also the Channel Impulse Response (CIR): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='𝑓 (𝑡) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='𝑁 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='𝑎𝑛𝛿(𝑡 − 𝜏𝑡) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Difference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Amplitude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='CSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Riderdistancefromthecar(meters) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 30AmplitudeDifference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier: 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='900 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Packet numbers() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='First multipath ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='15000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Second multipath ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='12500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='7500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Rider distance from the car (meters)7000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Firstmultipath ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Second multipath ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Rider distance from the car (meters)(a) Power delay profile of when the rider is LoS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='. (b) Power delay profile of when the rider is nLoS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 6: Power delay profile in LoS and nLoS conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' where 𝑎𝑛 and 𝑁 is the same as in Equation 1 and 𝛿(·) is the delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' By calculating the norm ∥𝑓 (𝑡)∥2 of the Channel Impulse Response 𝑓 (𝑡), we can get the Power Delay Profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Each of the signal samples in the Channel Impulse Response correlates to different multipath as their time to travel from the transmitter to the receiver differs due to differences in the traveled length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' By considering the IFFT theory, the time resolution Δ𝜏 is related to the sampling resolution Δ𝑓 mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' While increasing the number of bins in IFFT, the actual resolution does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' As such, we set our IFFT bins to the number of subcarriers, which is also the frequency sampling resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' For our collected data, 30 subcarriers are reported for each antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' By using two antennas, we obtain 60 PDP values as features per Wi-Fi packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We show the PDP values from one antenna in LoS condition in Figure 6(a), and in nLoS condition in Figure 6(b), where there are three people and two cars blocking the signal between the rider and his car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In both cases, the rider was on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We see how the PDP values are changing as the car approaches the rider from the left side of the X-axis and passes him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The PDP values do not necessarily tell if the rider is on the left or right side but help contextualize the packets of similar distance, and in LoS/nLoS conditions to provide additional information to the classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 5 CLASSIFICATION We consider different classifiers to classify the side of the rider (left vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' right), including k-Nearest Neighbor (kNN), Decision Tree (DT), and Support Vector Machine (SVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In addition, we design a Long Short-Term Memory (LSTM) neural network classifier by effectively integrating all the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Now, we describe the design of an LSTM and how we encode the relevant contextual and motion-related features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' As the vehicle approaches the rider, the motion of the vehicle, as well as the distance between the transmitter (the phone held by the rider) and the receiver (Wi-Fi receiver on the vehicle), provides additional features in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' For example, Wi-Fi signal differences between different antennas can vary across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' There are also Wi-Fi signal differences on the same antenna with different transmitter and receiver distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Unlike neural network architec- tures such as Fully-Connected Neural Network and Convolutional Neural Network (CNN), LSTM can better encode time series data with its feedback connections to remember values over arbitrary time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Thus it can exploit the temporal features introduced by the vehicle’s motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' While traditional classifiers like k-NN, DT, and SVM can capture features at a single time step, they lack the ability to take into account the temporal features as the signal is coming from either left or right in both LoS and nLoS situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The general execution of LSTM is described in equations below: 𝑖𝑡 = 𝜎(𝑊𝑖𝑖𝑥𝑡 + 𝑏𝑖𝑖 +𝑊ℎ𝑖ℎ𝑡−1 + 𝑏ℎ𝑖) (3) 𝑓𝑡 = 𝜎(𝑊𝑖𝑓 𝑥𝑡 + 𝑏𝑖𝑓 +𝑊ℎ𝑓 ℎ𝑡−1 + 𝑏ℎ𝑓 ) (4) 𝑔𝑡 = 𝑡𝑎𝑛ℎ(𝑊𝑖𝑔𝑥𝑡 + 𝑏𝑖𝑔 +𝑊ℎ𝑔ℎ𝑡−1 + 𝑏ℎ𝑔) (5) 𝑜𝑡 = 𝜎(𝑊𝑖𝑜𝑥𝑡 + 𝑏𝑖𝑜 +𝑊ℎ𝑜ℎ(𝑡 − 1) + 𝑏ℎ𝑜) (6) 𝑐𝑡 = 𝑓𝑡 ⊙ 𝑐𝑡−1 + 𝑖𝑡 ⊙ 𝑔𝑡 (7) ℎ𝑡 = 𝑜𝑡 ⊙ 𝑡𝑎𝑛ℎ(𝑐𝑡) (8) The main advantage of LSTM to other neural networks in tem- poral feature understanding is the memory cell 𝑐𝑡, which is used to accumulate state information in each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' To decide what to remember and forgot, Equation 3 and 4 calculate the input gate and forget gate value, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The input gate 𝑖𝑡 decides which information (which is calculated by Equation 5) is saved to the memory cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' On the other hand, the forget gate 𝑓𝑡 control which part of the previous cell status could be forgotten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' With these calcu- lations, we determine what is the new memory cell status through Equation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Additionally, how the memory cell 𝑐𝑡 propagates to the final state or output ℎ𝑡 (through Equation 8) is controlled by the output gate 𝑜𝑡 (calculated at Equation 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' This design allows the LSTM to take into account previous state information and can be self-learned through the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In these equations, 𝑥𝑡 is the data at time step 𝑡, 𝑏 is the bias in each network connection, the upper case 𝑊𝑖 and 𝑊ℎ represents the matrices of the weight of the input data and recurrent connection, respectively, and ⊙ is the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The architecture of LSTM is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' It has an input size of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 3 LSTM layers are stacked with 256 hidden units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' They are followed by a linear layer with an input size of 256 and an output size of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' A Softmax layer is added after the linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The training uses a cyclic learning rate with a 5e-4 initial learning rate with maximum epochs of 650 with the patience of 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' For loss function, we use cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We have a dropout of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 in the LSTM layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The sequence length in LSTM for each sample (or, a window) needs to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, we observe burst of packet losses in nLoS conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' As a result, the number of packets varies from windows to windows (in 3 seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Hence, the length of the LSTM sequence needs to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We take the median of the number of packets of the windows of the training set, which is 855 packets and set that the sequence length of LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' If there are more packets, 30 Power Delay Profile 8 52050 6 4 5 17-15 14-10 23 6 9 Rider distance from the car (meters)30 525 Power Delay Profile 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='0 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 0 19 17-14-12-10 0 2 61113 17 Rider distance from the car (meters)LSTM 1 (256) LSTM 2 (256) LSTM 3 (256) … 𝑍� 𝑍� FC (256) Log softmax log � exp �𝑍�� ∑ exp �𝑍�� � � 1/0 Classification Result Amplitude Diff (N) PDP (M) Multipath (1) FC (2) Figure 7: LSTM architecture then we ignore the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' If there are fewer packets, then we perform zero padding at the end of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In this way, we actually take 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 seconds of Wi-Fi packets half of the time for the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Before feeding the CSI amplitude difference, power delay profile, and multipath profile features to LSTM, we normalize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' This is important to make sure different features with different scales (especially the dominant multipath) do not force the network to weigh differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' So, the features from the multipath profile and power delay profile need to be crafted in a way that even after nor- malization, the distinction of LoS and nLoS does not disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In order to ensure that, we create just one feature using the multipath profile by dividing the magnitude of the dominant multipath with that of the less dominant multipath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' For the power delay profile, adding 60 input channels to LSTM may cause over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' So, we apply Principal Component Analysis of the 60 PDP features and take the top M principal components to feed to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We vary M from 3 to 5 and show the results in the Evaluation sec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' After this process, the normalization retains the LoS and nLoS distinction and reduces the number of input channels to LSTM to reduce overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 6 DATA COLLECTION 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='1 System Setup To extract the Wi-Fi data in an automotive environment, we utilize a laptop with an Intel 5300 Wi-Fi Network Interface Card (NIC) for portability, as shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We use the Linux CSI tool [12] to collect PHY layer CSI information from received Wi-Fi packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The car is driven with this set up for receiving Wi-Fi packets about 10-20 miles per hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The three antennas are placed in the dashboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We mark them as A (leftmost), B (middle), and C (rightmost), when viewed from inside of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Although we collect data with 3 antennas, we use only two antennas for our approach (antennas A and C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' On the rider side, the rider stands with a Pixel 2 XL phone that serves as an Access Point (AP) at 5 GHz to which the laptop is connected to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' An Android app from the phone generates Wi-Fi traffic by pinging the laptop as shown in Figure 9(a), that we Table 1: Distribution of 85 rides under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Rider left side Rider ride side Only Rider 7 6 People both sides of the rider (no car) 5 6 Two other people blocking signal 13 14 Two cars blocking signal (no other people) 10 12 Two cars and three other people blocking signal 6 6 developed and can achieve a packet transmission rate of up to 350 packets per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2 Collected Dataset In order to consider realistic scenarios with LoS and nLoS condi- tions, we collect data of 85 rides under five different conditions: (a) only rider standing, (b) people standing on both sides of the rider, (c) two other people blocking the signal, (d) two other parked cars blocking the signal, and (e) two other cars and three other people blocking the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We collect data when the rider is on the left and right sides of the car in all these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Table 1 shows the number of rides under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 10 shows a drone image of the case (e), where the rider is standing, and three people and two cars are blocking the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The Wi-Fi receiving unit is in the blue car that was being driven from the left to the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We split the dataset into training (60%), validation (20%), and testing (20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We do this per for each condition and for each side of the rider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' For example, when the rider is at the left side and two cars blocking the signal, we have 10 such rides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We take CSI data of 6, 2, and 2 rides for training, validation, and testing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In that way, the test set has data of disjoint rides and under all conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' For each ride, we split the sequence of CSI values into a 3 seconds window with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='4 seconds stride length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' This gives us 1032 windows for training, 286 windows for validation, and 285 windows for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='3 Ground Truth Collection In order to collect the ground truth of whether the rider is at the left or right side of the car, one can just record the timestamps of received packets for each side of the rider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, we would like to collect the (x,y) location of the car when each packet was received to have a better understanding of how the CSI changes when the vehicle approaches the rider and leaves the rider at each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In order to achieve this goal, we use an off-the-shelf consumer drone hovering above the data collection site to record the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' An example frame from the recorded video is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Before the data collection, we first determine landmark locations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', the rider’s location) and four positions that can form a rectangle area with tape-measured ground truth coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Next, we place solid red-colored papers at each location and on top of the car to enable simple color-based pixel tracking through color thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In the recorded video, we use the four locations to perform Homography transformation so that the pixel plane and real-world plane are parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' This transformation creates a straightforward translation from the pixel coordination system to the real-world coordination system through scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Then we can track the vehicle in the pixel domain and interpolate the real-world (x,y) location through the @ BOSCHFigure 8: In-vehicle system setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The left picture shows the antenna viewed from outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The middle picture shows Wi-Fi antennas placed on the central dashboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Antennas from the left to the right are labeled as 𝐴, 𝐵, and 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The right picture shows a laptop with an Intel 5300 NIC and connected with antennas through cables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' (a) Android App developed to gener- ate Wi-Fi traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' (b) View of how the phone is held rel- ative to the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 9: Data collection equipment and environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Prior to each data collection, we also time-synchronize the Android phone, the laptop with the Intel Wi-Fi chipset, and the drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The time-synchronization between the phone and drone is achieved by capturing the phone’s time with millisecond accuracy at the beginning of each drone’s video;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' thus, we can calculate the timestamp based on the frame rate and a reference frame that has the phone’s time clearly recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We also capture a screenshot with both the phone’s time (through the laptop’s camera) and the laptop’s time displayed with millisecond accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' thus, the time difference between them can be easily calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We apply these time offsets to change the timestamp recorded on the laptop and the drone to match the time on the phone for time synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 7 EVALUATION In this section, we estimate the accuracy of CarFi and compare it with state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We investigate the effect of antenna spacing, subcarrier selection, and window size on the performance of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Also, we estimate its execution time, and range in both LoS and nLoS conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='1 Accuracy To the best of our knowledge, there is no state-of-the-art Wi-Fi- based rider side determination technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' So, we implement a few Figure 10: Drone image of three people and two vehicles in between the Wi-Fi transmitter and receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' baseline methods to compare with our approach in terms of accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Baseline 1: CSI phase difference based approach: Although our approach does not require phase calibration, in order to inves- tigate and compare with a phase difference based approach, we perform phase calibration of the antenna chains of the Intel 5300 chipset attached with the laptop using a method similar to [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We use another laptop with Intel 5300 chipset to transmit Wi-Fi packets through an RF splitter, where all CarFi three receiver antennas are connected to the RF splitter’s output as shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' These three antennas should receive the Wi-Fi signal at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, due to the slight path distance difference within the RF splitter, we switch the receiver antennas’ connecting locations and record the phase information in each connection combination to eliminate the difference introduced by the RF splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' By removing the offset we measured, we correct the antenna phase offset in our collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The system also introduces Sampling Time Offset (STO) and Sampling Frequency Offset (SFO) as the sampling clocks and frequencies are unsynchronized between the receiver and the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We follow SignFi [25] to remove STO and SFO through multiple linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We use only antennas 𝐴 and𝐶, and estimate the phase difference by subtracting unwrapped phase𝐴 from unwrapped phase𝐶 of each window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Ideally, the phase difference should be positive (negative) when the rider is on the left (right) side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' But for 30 different sub- carriers, the patterns vary significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' As an example, we show the phase difference when the rider is at the right side in LoS condition and when he was blocked by two cars in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 11: Antenna chain phase calibration of the Intel 5300 chipset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' (a) Unwrapped phase difference of antennas (𝐶 - 𝐴) in LoS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' (b) Unwrapped phase difference of antennas (𝐶 - 𝐴) in nLoS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 12: Unwrapped phase difference in LoS and nLoS conditions when the rider is in the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 13: Positive and negative votes of unwrapped phase differ- ence between antennas when the rider is in the both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Figure 14: Estimating effective phase difference between antennas (𝐶 - 𝐴) when the rider is at the right side in LoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Since the unwrapped phase difference change over time, we consider four different ways to compute the features to capture phase difference between antennas: (a) We average all phase differences of all sub-carriers of all packets within a window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The intuition is that mean of phase difference should be different for different sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' (b) Similar to (a), but instead of all the sub-carriers, we use just the first sub-carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' (c) We divide the window into a few sub-windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The reason for sub-windowing is to reduce the propagation error of phase unwrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Then, we average all phase differences of all sub-carriers in each sub-window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We remove 20% sub- windows with large variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Then, we compute a positive or negative vote for each sub-window based on the sign of its phase difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We count the numbers of positive and negative votes, and use them as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We plot the num- ber of positive and negative votes for all the 1032 training windows and plot them in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The green line shows when the positive and negative votes are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' It shows that when the rider is on the left (right) side, there are more positive (negative) votes (as they should be).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We calculate such votes for all the sub-carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' (d) After sub-windowing, we compute an effective phase differ- ence for each subcarrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The intuition is that phase differ- ence should be stable for each subcarrier because the central frequency is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We choose an effective phase differ- ence that covers most phase differences within two radians and has the smallest mean error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' An example of such an effective phase difference is shown in Figure 14 when the rider is on the right side and in LoS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' It shows that even though the phase difference fluctuates, the effective phase difference is negative (as it should be since the rider is on the right side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We estimate like this for 30 sub-carriers and use all 30 effective phase differences as features to the classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We feed these features to kNN, DT, and SVM classifiers and show the results of rider side classification in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' For kNN, we vary the value of k from 3 to 15 and report the accuracy with the best k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We see the highest accuracy we get from the phase difference based approach is only 56%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Baseline 2: RSS difference based approach: When we collect data, we also collect RSS (Received Signal Strength) values from each antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We feed the average RSS difference of antennas (𝐶 - 𝐴) of each window to different classifiers, including KNN, Decision tree, and SVM, to classify the rider side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The results show that the highest accuracy is 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='6% that came from both K-NN and SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' It provides higher accuracy than the CSI phase difference based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Baseline 3: CSI amplitude difference based approach: Since the CSI amplitude difference changes over time, we con- sider different ways to compute features to capture amplitude dif- ference of antennas (𝐶 - 𝐴): (a) We average all CSI amplitude differences of all sub-carriers of all packets within a window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' (b) Similar to (a), but we use only the first sub-carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' (c) Similar to (a), but we also add average RSS difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' (d) Similar to (b), but we also add average RSS difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We feed the features to kNN, DT, and SVM classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Its shows the highest accuracy is 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' when ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='CSI phase difference (rad) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='Subcarrier 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Baseline KNN (%) DT(%) SVM(%) 1(a) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2 (k=5) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='4 1(b) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='4 (k=11) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2 1(c) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='6 (k=3) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='6 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='6 1(d) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='6 (k=3) 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='6 (k=10) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='6 3(a) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2 (k=5) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='6 3(b) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='1 (k=3) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='9 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='7 3(c) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 (k=7) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='3 3(d) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='3 (k=8) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 combining the average CSI amplitude difference and average RSS difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We also implement our LSTM based network and change net- work parameters, including the size of hidden dimensions and number of layers to see how that affects performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The results are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' It shows that when we use our variance based subcarrier selection, the accuracy is higher than when all sub- carriers are used, or only the first subcarrier is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We see that we get 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='44% accuracy when we combine variance based subcarrier selection, power delay profile, and multipath profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' This highest accuracy came from when we select 14 subcarriers with VbSS, ob- tain 3 PDP features and 1 multipath profile feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' If we feed the exact same features to kNN, DT, and SVM, we get 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='4%, 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5% , 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2% accuracy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Hence, our LSTM based architecture increases accuracy by 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='24% from exactly the same input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2 Sensitivity Analysis In this section, we analyze the effect of antenna spacing, subcarrier selection, and window size on CarFi performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Effect of antenna spacing: In our analysis, the default antenna spacing was 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2 cm, which produced 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='44% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Since we collected data with three antennas, we can use antenna 𝐴 and 𝐵 to see how the performance looks like when the antenna spacing is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='6 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We keep the best performing network’s parameter the same and run the experiment with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='6 cm spacing and find the accuracy is only 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='79%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Thus, increasing antenna spacing helps improving accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Effect of window size: In our analysis, the default window size is 3 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We keep the best performing network’s parameter the same and run the experiment by varying window sizes to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5, 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5, 2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5, and 3 seconds find the accuracy is 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='95%, 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='36%, 80%, 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='17%, 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='03%, 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='44%, respectively as shown in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We see that longer windows provide higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Effect of number of sub-carriers: We keep the best perform- ing network’s parameter the same and run the experiment with changing the number of sub-carriers from 1 to 16 and show the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='0 Window Size (seconds) 50 60 70 80 90 100 Accuracy (%) Figure 15: Effects of window size on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 16 Number of subcarriers 86 88 90 92 94 96 Accuracy (%) Figure 16: Effects of number of sub-carriers on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' impact of the number of sub-carriers through our VbSS method on accuracy in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' When we choose only one subcarrier, we choose subcarrier 1, as it provides 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='47% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We achieve the highest accuracy (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='44%) when the number of sub-carriers is 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='3 Execution Time We train our LSTM using Nvidia GeForce GTX 1080 Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' It takes about two hours to train the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, the infer- ence is rapid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We estimate how long it takes to perform inference in a powerful GPU like NVidia GeForce GTX 1080 Ti as well as an embedded GPU like Nvidia Jetson Nano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' It takes only 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='77 and 850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='37 milliseconds to execute the inference process in 1080 Ti and Jetson Nano, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Hence, the solution can be run on embedded GPUs in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Also, there are several ways to optimize (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', recompiling with TensorRT can significantly reduce inference time on Jetson devices) and prune the model to compress the network, which will reduce inference time [53] [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We leave this to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='4 Range Analysis In this section, we estimate how far CarFi can operate in both LoS and nLoS conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We collect additional data for this evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We have a person standing at different distances ranging from 10 meters to 120 meters in front of the car in both LoS and nLoS conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We transmit 10,003 packets from each location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' To create a nLoS condition, we have a person standing between the phone and Wi-Fi receiving unit placed in the car dashboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The Packet Delivery Ratio (PDR) at different distances from the car is shown in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We see that even at a range of 120 meters, the PDR is 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='30% in LoS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, in nLoS situation, the PDR drops sharply to 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='65% in 70 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' At 120 meters, the PDR is Table 3: Results of left vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' right classification when LSTM with different features are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Description Input Dim Hidden Dim Number of layers Optimizer Accuracy Variance-based Subcarrier Selection (VbSS) 12 256 3 RMSProp 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='33% Variance-based Subcarrier Selection (VbSS) 12 256 3 Adam 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='57% Variance-based Subcarrier Selection (VbSS) 12 128 3 RMSProp 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='28% Variance-based Subcarrier Selection (VbSS) 12 256 4 RMSProp 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='82% Variance-based Subcarrier Selection (VbSS) 12 128 4 RMSProp 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='63% First Subcarrier only 1 256 3 RMSProp 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='47% All sub-carriers 30 256 3 RMSProp 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='47% VbSS + Multipath Profile (MP) 12+1 256 3 RMSProp 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='33% VbSS + Power Delay Profile (PDP) 12+5 256 3 RMSProp 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='33% VbSS + Power Delay Profile (PDP) 12+3 256 3 RMSProp 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='68% VbSS + PDP + MP 12+3+1 256 3 RMSProp 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='08% VbSS + PDP + MP 14+3+1 256 3 RMSProp 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='44% 10 20 30 40 50 60 70 80 90 100 110 120 Rider distance from the car (meters) 0 20 40 60 80 100 Packet Delivery Ratio (%) LoS nLoS Figure 17: Packet delivery ratio at different distances from the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' So, we see that in LoS conditions, CarFi will operate beyond 120 meters range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 8 DISCUSSION 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='1 Generalizability Although the data was collected from one large parking lot, we put an effort to introduce variation in the rides by asking the volunteers to stand differently to block the signal, move while blocking the signal, drive at different speeds, and vary the speed in different rides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' As a result, there is a significant variation in the dataset, and we expect the model to generalize to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' One particular reason we were not able to collect data from a busy street is that the Wi-Fi of the laptop needed to stay connected to the phone for data collection with CSI Tool [13], which is very difficult to obtain in busy streets as the car can easily go out of the Wi-Fi range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Currently, we are switching to Nexmon framework [37] for collecting CSI data, where the phone will inject packets at a particular Wi-Fi channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' This will allow us to perform a large-scale data collection from busy streets for testing the generalizability of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We leave it to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2 Limitations One potential limitation of our approach is that if someone else books the ride on behalf of the rider and the rider has a different phone, then the proposed solution may not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' If the rider does not co-operate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', by disabling Wi-Fi), then it will not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Also, it will need support from the rider-hailing service providers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', Uber, Lyft) to enable this service into their apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, they can incorporate such a service using their apps and dashboard products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Also, our approach assumes that the rider is waiting for the car on a side of a street while the car is moving towards or away from the rider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' If the rider books the trip from inside, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', from a restaurant, and waits inside for the car to arrive, then we will not be able to leverage the motion related features as effectively, and it may not be able to assist the driver with the rider side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, when the rider comes out of the restaurant, we can ask him to walk along his side of the street to help us capture some features to determine his side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Due to the time sensitive nature of the application, assuming it takes maximum 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='85 seconds (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='85 seconds for execution and maximum 3 seconds of window, although 50% time we only need 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 seconds to collect enough packets and perform classification) to run CarFi on a Jetson Nano and Wi-Fi range of 70 meters in nLoS, CarFi can only determine the rider side when the rider is in front of the car if the vehicle approaches towards the rider at 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='67 miles per hour or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, it can work up to 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='34 miles per hour vehicular speed if we are not required to determine rider side while the rider is in front of the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='3 Lessons Learned and Future Work Our exploratory study shows that the phase difference between antennas in an automotive environment is very noisy and does not provide as high accuracy as amplitude difference-based methods for determining rider side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We also find that increasing the antenna distance even by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='6 cm helps improving accuracy for the amplitude- based approach significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The maximum length of an object that can be placed in a dashboard is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='5 inches by the laws of California, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We plan to increase antenna spacing and see if we can achieve higher accuracy at smaller window sizes within this constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' We leave this to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Also, we estimate the rider side independently from each window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In the future, we can take into account all the packets of the past windows from the same ride, assuming that the rider did not cross the street while the car is approaching and have a bigger window for rider side determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Also, we plan to collect data from busy streets and evaluate the performance in challenging scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 9 RELATED WORK 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='1 Human Detection and Identification There has been a significant amount of work in detecting the pres- ence of humans and identifying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' For the ride-hailing appli- cation, the vehicle needs to detect the presence of the rider and identify him/her before localizing him/her.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Humans can be detected using cameras [48, 54], depth sensors [10, 11, 20, 21, 26, 28–30], IR- array sensors [27], mmWave radars [5, 52], Wi-Fi [4, 7–9], and using other sensors for various purposes [14, 33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Vision-based systems such as [39, 43, 44] can be used to identify the person but requires prior knowledge such as the facial features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' These systems are privacy invasive (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=', facial recognition systems are being banned in multiple cities for privacy concerns) and gener- ally do not work if the subject is at a distance and can be occluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Systems such as ID-Match [19] uses RFID tags and a 3D depth cam- era, FORK [28] uses a depth sensor, and [23] uses RFID and BLE to identify individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' These works require additional sensors or devices, which are not practical in the ride-hailing scenario, difficult to scale, and potentially costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Similar to EyeFi [7] and RFCam [4], we utilize the phone as an identifier as it is already being utilized in the ride-hailing application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='2 Localization Numerous works have focused on localization using Wi-Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' [18] exploits Wi-Fi subcarrier CSI values to create virtual antennas to obtain higher resolution Angle of Arrival (AoA) estimation, then using multiple Wi-Fi receivers to triangulate the transmitter for localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The usage of multiple Wi-Fi receivers is expensive, and AoA estimation is unstable in our automotive environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' [49] proposes a higher resolution of power delay profile by utilizing CSI splicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' This method requires a special Wi-Fi configuration, which is impractical for our application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, it does inspire us to use power delay profile as one of our features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Other works such as [38, 41, 42, 45] have examined using time-of-flight, frequency, and multipath to estimate AoA and triangulate the locations of the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' These methods are more suited for indoor settings and more prone to environmental changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Using simple features such as RSSI has been explored in previ- ous works to perform vehicle localization [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Similarly, [31] uses a fingerprinting method to localize vehicles in a car park.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' While fingerprinting is simple and easy to perform, they are prone to en- vironmental changes and lacks generalization ability as it requires prior knowledge for each place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' [51] locates a bus by scanning Wi-Fi APs surrounding the bus route and predicts the time for the bus’s arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, it still lacks the generalization ability and can not provide fine-grained location information regarding the rider side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' One work that is closely related to our work is [15], which uses Wi-Fi Fine Time Measurement (FTM) to measure the distance (through ToF) and achieve high precision localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, is- sues such as the time needed to perform such measurement (which can take several seconds), the requirement of the phone to con- nect, and the need to place antennas on both sides of the vehicle’s roof prevent it from being conveniently deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Similarly, [32] utilizes both CSI and FTM to achieve higher accuracy with added AoA (Angle of Arrival) and AoD (Angle of Departure) measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, this method still depends on FTM that most of the smartphones do not support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='3 Wi-Fi Sensing While some work in Wi-Fi sensing does not directly apply to local- ization problems, they can provide meaningful features that may be utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' For example, [47] use Wi-Fi CSI phase change dynamics with Fresnel zone model to determine the human subject’s walk- ing direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The phase change dynamics contain environmental changes around the transmitter and receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In [16], the author demonstrated how the Wi-Fi signal contains detailed information that can be used to reconstruct human pose, and a neural network can be used to extract deeper features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' In these works, the transmit- ters and receivers are stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' However, in our case, the receiver is moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 10 CONCLUSION In this paper, we investigate the feasibility of using Wi-Fi based street side determination of riders from a car to assist drivers to locate their riders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' This method can potentially enable a smoother pick-up experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' Our approach uses a two-antenna Wi-Fi chipset for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' After extracting CSI values, it computes relevant features by leveraging the motion of the car and utilizes a data- driven technique to determine the rider side on an embedded GPU in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' By performing extensive evaluation in the real-world in both LoS and nLoS conditions, we see CarFi achieves 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content='44% accuracy for estimating the rider side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' The approach can potentially be very useful when self-driving cars and robotaxis hit the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfoP0g/content/2301.01592v1.pdf'} +page_content=' 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b/htE_T4oBgHgl3EQf3xxC/content/tmp_files/2301.08348v1.pdf.txt @@ -0,0 +1,229 @@ +arXiv:2301.08348v1 [cs.CC] 19 Jan 2023 +A Quantum EL Theorem +Samuel Epstein∗ +January 23, 2023 +Abstract +In this paper, we prove a quantum version of the EL Theorem. It states that non-exotic +projections of large rank must have simple quantum states in their images. A consequence to +this is there is no way to communicate a quantum source with corresponding large enough von +Neumann entropy without using simple quantum states. +1 +Introduction +Quantum information theory studies the limits of communicating through quantum channels. In +Holevo (1973), the Holevo bound was proven, providing an upper bound on the amount of classical +information shared between two parties that can prepare and measure mixed states. The Holevo +bound states that only n bits of classical information can be accessed from n qubits. Schumacher’s +theorem Schumacher (1995) gives necessary and sufficient conditions under which there exists a +reliable compression scheme to compress and decompress a quantum message with high fidelity. +There is a large literature about the potential of quantum algorithms, with the most famous +being Shor’s factoring algorithm. +There exists a relatively new area combining algorithms and +quantum mechanics: the intersection of Algorithmic Information Theory (AIT) and Quantum +Information Theory. There are several interesting results in this new field. For example, in Epstein +(2021b), it was shown that given a quantum measurement (i.e. POVM) when it is applied to a +pure quantum state, the vast majority of outcomes is meaningless random noise. +This research program involves finding the quantum equivalent to definitions and theorems in +AIT, with the primary concept being an quantum version of Kolmogorov complexity K(x). There +are several such definitions that measure the algorithmic information content in a mixed or pure +quantum state. In this paper we will use the definition K(|ψ⟩) in Vitanyi (2000), which says a pure +state |ψ⟩ is complex if there is no simple (in terms of its classical enoding) pure state that has high +quantum fidelity with |ψ⟩. The results of this paper also applies to quantum algorithmic entropy, +G´acs (2001). In Epstein (2019), the quantum equivalent to algorithmic information and random +deficiencies were defined. In addition conservation inequalities were proven with respect to unitary +transform +In this paper we prove a Quantum EL Theorem. In AIT, the EL Theorem Levin (2016); Epstein +(2019) states that sets of strings that contain no simple member will have high mutual information +with the halting sequence. It has many applications, including that all sampling methods produce +outliers Epstein (2021a). The Quantim EL Theorem states that non exotic projections of large +rank must have simple quantum pure states in their images. By non exotic, we mean the coding +of the projection has low information with the halting sequence. +∗samepst@jptheorygroup.org +1 + +The Quantum EL Theorem can be used to address open issues in Quantum Information Theory. +In G´acs (2001) the following remark was made. +Remark. (G´acs (2001)). Maybe the study of the problem for quantum description complexity helps +with the understanding of the problem for von Neumann entropy, and its relation to coding tasks +of quantum information theory. +The theorem in this paper helps address this remark. +Claim. As the von Neumann entropy associated with the quantum source increases, the lossless +quantum coding projectors have larger rank and thus must have simpler (in the algorithmic quantum +complexity sense) pure states in their images. +2 +Related Work +For information about the history and foundation of algorithmic information theory, we refer readers +to the textbooks Downey and Hirschfeldt (2010) and Li and Vit´anyi (2008). +There are several +definitions that model the algorithmic content of a quantum state. In Berthiaume et al. (2001), +the complexity of a quantum state is equal to the size of the smallest quantum Turing machine +that can approximate the state to a given fidelity. In Mora and Briegel (2005), the algorithmic +complexity of a quantum state is equal to the minimal length of an encoding of the preparation +of the state through quantum gates. In G´acs (2001), the algorithmic entropy of a quantum state +is measured by the negative logarithmic of the state multiplied by a universal lower computable +semi-density matrix. +In Vitanyi (2000), the entropy of a pure quantum state is equal to the +classical complexity of an elementary approximating state plus the negative logarithm of their +fidelity. A quantum version of Brudno’s theorem was proven in Benatti et al. (2006). Randomness +for infinite quantum spin chains, called quantum Martin L¨of random sequences, was introduced in +Nies and Scholz (2019). An infinite version of algorithmic entropy can be found at Benatti et al. +(2014). +3 +Conventions +The length of a string x ∈ {0, 1}∗ is ∥x∥. For positive real function f, <+ f, >+ f, and =+ f +is used to represent < f + O(1), > f + O(1), and = f ± O(1). The encoding of x ∈ {0, 1}∗ is +1∥x∥0x. For the nonnegative real function f, the terms logf, and =logf represent the terms +f−O(log(f+1)), and =f±O(log(f+1)), respectively. +For strings x, y ∈ {0, 1}∗, the output of algorithm T on input x and auxilliary input y is denoted +Ty(x). An algorithm T is prefix free iff for strings x, y, s ∈ {0, 1}∗, ̸= ∅, if Ty(x) halts then Ty(xs) +does not halt. There exists a universal prefix free algorithm U, where for all prefix-free algorithms +T, there exists a t ∈ {0, 1}∗, where for all x, y ∈ {0, 1}∗, Uy(tx) = T(x). +This U is used to +define Kolmogorov complexity, with K(x|y) = min{∥p∥ : Uy(x) = p}. The universal probability +of x ∈ {0, 1}∗, conditional to y ∈ {0, 1}∗, is m(x|y) = �{2−∥p∥ : Uy(p) = x}. By the coding +theorem, we have − log m(x|y) =+ K(x|y). We use I(x; H) = K(x) − K(x|H) to be the amount of +information that the halting sequence H ∈ {0, 1}∞ has about x ∈ {0, 1}∗. +2 + +We use Hn to denote a Hilbert space with n dimensions, spanned by bases |β1⟩ , . . . , |βn⟩. A +qubit is a unit vector in the Hilbert space H2, spanned by vectors |0⟩, |1⟩. To model n qubits, we +use a unit vector in H2n, spanned by basis vectors |x⟩, where x is a string of size n. +A pure quantum state |ψ⟩ of length n is a unit vector in H2n. Its corresponding element in the +dual space is denoted by ⟨φ|. The conjugate transpose of a a matrix A is A∗. Tr is used to denote +the trace of a matrix. Projection matrices are Hermitian matrices with eigenvalues in {0, 1}. +Pure quantum states are elementary if their values are complex numbers with rational coeffi- +cients, and thus they can be represented with finite strings. Thus elementary quantum states |φ⟩ +can be enncoded as strings, ⟨|φ⟩⟩ and assigned Kolmogorov complexities K(|φ⟩) and algorithmic +probabilities m(|φ⟩). They are equal the complexity (and algorithmic probability) of the strings +that encodes the states. More generally, a complex matrix A is elementary if its entries are complex +numbers with rational coefficients and can be encoded as ⟨A⟩, and has a Kolmogorov complexity +K(A) and algorithmic probability m(A). +4 +Quantum Projections +Simplicity is measured according to the classical information content of a pure state. It is similar +to the definition in Vitanyi (2000) except a classical Turing machine is used instead of a quantum +Turing machine. +Definition 1 (Complexity of a Quantum Pure State). +For n qubit state |φ⟩, H(|φ⟩) = min{K(|ψ⟩) − log | ⟨φ|ψ⟩ |2 : |ψ⟩ is an elementary pure state}. +A probability is elementary if it has finite support and all its values are rational. The deficiency +of randomness of a string x with respect to an elementary probability mesaure Q is d(x|Q) = +⌊− log Q(x)⌋ − K(x|⟨Q⟩). The stochasticity of a string is Ks(x) = minQ{K(Q) + 3 log d(x|Q)}. +Lemma 1 (Epstein (2021a); Levin (2016)). Ks(x) 2m. Then, relativized to (n, m), min|φ⟩∈Image(P ) H(|φ⟩) 2m−n−1 otherwise one can create a Q-expectation test, t, such that t(R) = exp d. This +is a contradiction because +1.44d <+ log(P) <+ d(P|Q) <+ d, +for large enough d which we can assume without loss of generality. Thus there exists i such that +⟨ψi| P |ψi⟩ ≥ 2m−n−1. +Thus |φ⟩ = P |ψi⟩ / +� +⟨ψi| P |ψi⟩ is in the image of P and | ⟨ψi|φ⟩ |2 = +⟨ψi| P |ψi⟩ ≥ 2m−n−1. The elementary state |ψi⟩ has classical Kolmogorov complexity K(|ψi⟩) 2m. Then, +relativized to (n, m), min|φ⟩∈Image(P ) H(|φ⟩) 103 cm−3 (see Brinks 1990), +and assuming it to be similar in NGC 2639, we can estimate the CO +gas pressure as 𝑃 = 𝑛𝑘𝐵𝑇, where 𝑘𝐵 is the Boltzmann constant. For +the CO gas ring radius of 3 kpc, the volume of the disk-like cavity is +1.25 × 1066 cm3 and the 𝑃𝑉 work done is > 3.44 × 1054 erg. +We can also estimate the jet mechanical powers using the relations +derived for low luminosity AGN by Merloni & Heinz (2007). Using +only the lobe flux densities at 5 GHz, we find that the jet power for +the east-west VLA lobes is 7.7 × 1042 erg s−1 and the time-averaged +power (for a spectral age of 2.8 Myr) is 6.8×1056 erg. Similarly, for the +north-south VLA lobes, the 5.5 GHz jet power is 1.3 × 1042 erg s−1, +and its time-averaged power (over 12 Myr) is 4.9 × 1056 erg. Finally, +using an average lobe spectral index of −0.3, the jet power for the +north-east - south-west uGMRT lobes is 5.7 × 1042 erg s−1, and its +time-averaged power (over 34 Myr) is 6.1×1057 erg. Therefore, only +∼ 0.5% of the east-west jet power is sufficient to push back the CO +gas in NGC 2639; these numbers are ∼ 0.7% for the north-south jets +and ∼ 0.06% for the north-east - south-west jets, respectively. +4.3 Indications of Star-formation Quenching +The GALEX data on NGC 2639 shows a clear deficiency of far- and +near-UV (FUV and NUV) emission from the central ∼ 6 kpc of the +galaxy (see Figure 6). The NUV band directly traces stars formed +over the last 200 Myr and therefore probes recent star formation +(Kennicutt & Evans 2012). The central void in the GALEX NUV +image, therefore, is consistent with the suggestion of star-formation +quenching in the central few kpc in NGC 2639. We attempt to quantify +this further ahead. +The global Schmidt law for star-forming galaxies has been given +by Kennicutt (1998) as: +ΣSFR = (2.5 ± 0.7) × 10−4 +� +Σgas +1 M⊙ pc−2 +�1.4±0.15 +M⊙ yr−1 kpc−2 +(1) +where the SFR surface density, ΣSFR, can be derived from gas sur- +face density, Σgas. For NGC 2639, using Σgas ≡ ΣH2 = 21 M⊙ pc−2 +for a region of 4.8 kpc (Raluy et al. 1998), ΣSFR should be +0.0177 M⊙ yr−1 kpc−2. The SFR surface density can also be com- +puted from the SFR estimates using the following expression: +ΣSFR = +SFR +𝜋a2 +� +d +206265 +�2 +(2) +where the parameter 𝑎 corresponds to the semi-major axis of the +telescope aperture in arcsec and 𝑑 is the distance to the galaxy in Mpc +(Catalán-Torrecilla et al. 2015). Several estimates for SFR have been +derived for NGC 2639 in the literature. Table 3 provides the estimates +of ΣSFR obtained using different SFR tracers. The telescope details +for individual tracers are also provided. These estimates of ΣSFR are +suggesting that star-formation is quenched in NGC 2639 compared +to the global Schmidt law for star-forming galaxies by a factor of +5 − 18. This is fully consistent with the GALEX NUV image of +NGC 2639, showing a deficit in recent star-formation in the central +∼ 6 kpc region. +5 DISCUSSION +Sebastian et al. (2019) have argued that the multiple radio lobes +seen in NGC 2639 are due to minor mergers that did not disrupt the +morphology of the host galaxy. We also find that the line of sight +velocity map of the host galaxy from the CALIFA survey shows that +the stars of NGC 2639 are relatively undisturbed and are more or +less in uniform motion (de Amorim et al. 2017). The misaligned +jets are the result of new accretion disks formed from mergers, with +jet directions conserving the angular momentum of the inflowing +gas. This scenario is consistent with NGC 2639 having a large bulge +component surrounding the nucleus (Cox et al. 2007) as gravitational +forces and torques that result from mergers disrupt the orbital path +of stars causing randomised bulge orbits. Thus, if the minor merger +scenario were true, the spectral age results of the multiple lobes +indicate that minor mergers occurred every 9 − 22 Myr apart in the +last ∼ 30 Myr. +The expected minor merger rate for a galaxy like NGC 2639 (red- +shift of 0.01113 and stellar mass of 1.48 × 1011 M⊙) is ∼ 13 Myr +following the work of Conselice et al. (2022), who used observa- +tional data from the REFINE survey. We used their minor mergers +best fit line (with stellar mass ratios of 1:10) to obtain this estimate. +The estimate of ∼ 10 Myr also matches the estimates obtained via +theoretical studies as well as galaxy-merger simulations (Hopkins +et al. 2010; Capelo et al. 2015). It would therefore be fair to conclude +that at least three minor mergers have taken place in the lifetime of +NGC 2639. Each of these mergers may have resulted in the formation +of a new accretion disk with no memory of the previous accretion +disk direction, primarily driven by the angular momentum of the +infalling material itself (e.g., Kharb et al. 2006). Accretion through +these disks would have resulted in the several differently-oriented jet +episodes that are observed in NGC 2639. As noted in Section 4.2, +each of these jet episodes have sufficient mechanical power to dis- +place the CO molecular gas from the central few kpc of the host +galaxy. However, due to the directionality of the collimated jets, it +takes multiple differently-oriented jets to create the CO gas ring seen +in NGC 2639. +The early-type galaxy and LINER, NGC 1266, shows the presence +of a CO molecular outflow, no signatures of galaxy interactions, and +a possible radio jet at 1.4 GHz (Alatalo et al. 2011). There is also a +centrally concentrated molecular component which is different from +the case of NGC 2639, where instead, a central deficiency is ob- +served. Alatalo et al. (2011) suggest that (the sole) jet in NGC 1266 +is sufficient to drive the molecular outflow using only 2% of its total +power at 1.4 GHz. We note, however, that multi-resolution, multi- +frequency radio observations are required to truly rule out the exis- +tence of multiple jet episodes in NGC 1266. Nesvadba et al. (2021) +detected a CO(1-2) molecular gas ring through ALMA observations +in the nearby spiral galaxy J2345−0449 with large kpc-scale radio +jets. Interestingly, the inner radius of the CO gas ring corresponds to +MNRAS 000, 1–9 (2022) + +8 +V. Rao et al. +Table 3. Estimates of star formation rate in NGC 2639 using different tracers +SFR +Telescope +SFR indicator +Aperture +ΣSFR +Reference +(M⊙ yr−1) +(′′) +(M⊙ yr−1 kpc−2) +0.92 +Nickel 1.0 m telescope at Lick Observatory +H𝛼 +73.5 +0.00099 +(1) +1 +Spitzer Space Telescope +IR +40 +0.0036 +(2) +0.57 +Calar Alto 3.5 m telescope +H𝛼 +36 +0.0026 +(3) +References: (1) Theios et al. (2016) using the relation, SFR (M⊙ yr−1) = 5.37 × 10−42 LH𝛼 (erg s−1), where LH𝛼 = 1041.23 (erg s−1) for NGC 2639 +(2) Sebastian et al. (2019) using the CLUMPYDREAM code +(3) Catalán-Torrecilla et al. (2015) using the H𝛼 line luminosity from the CALIFA survey +4.2 × 2.2 kpc, very similar to what is observed in NGC 2639. More- +over, they find that the molecular gas outflow in J2345−0449 has a +kinetic energy of 1.3 × 1057 erg. Again, only a small fraction of the +jet kinetic power in J2345−0449 (and as it happens in NGC 2639) +can suffice to drive molecular gas outflows. It is worth noting that the +radio source in J2345−0449 is also a restarted double-double radio +galaxy. +The absence of CO(1-0) gas in the inner 6 kpc region of NGC 2639 +could indicate the presence of “negative AGN feedback” by the jet. +The relatively small values of ΣSFR observed in NGC 2639 com- +pared to the Schmidt law for star-forming galaxies suggest that star- +formation quenching is taking place in NGC 2639. Simulations of +Mukherjee et al. (2018); Meenakshi et al. (2022) have shown that +the jet-ISM coupling is sensitive to the relative orientation of the jet +w.r.t the gas disk, as well as the power and age of the jet. It is stronger +for the jets, which are oriented at ≥ 45◦ w.r.t the gas disk, young +(≤ 2 Myr), and highly powerful (≥ 1045 erg s−1). In NGC 2639, a +single jet episode may not meet all the above criteria. However, three +jet episodes together can result in an efficient coupling with the ISM. +6 CONCLUSIONS +The Seyfert galaxy NGC 2639 exhibits four episodes of AGN jet ac- +tivity as evidenced by 735 MHz, 5.5 GHz, and 8.3 GHz frequency +observations via the uGMRT, VLA and the VLBA telescopes, re- +spectively. Using the spectral ageing software BRATS, we derive the +ages of the three pairs of lobes to be respectively, 34+4 +−6 Myr, 11.8+1.7 +−1.4 +Myr, and 2.8+0.7 +−0.5 Myr, with the uGMRT lobes being the oldest (we +did not derive an age for the VLBA jet). Using the “on” and “off” +times of these jets/lobes, the AGN jet duty cycle in NGC 2639 is +∼ 60%. NGC 2639 also shows a deficiency of molecular gas in its +central ∼ 6 kpc region. Less than 1% of the jet mechanical power for +each of the jet episodes taken individually, is sufficient to move the +molecular gas. However, the creation of a ring in the molecular gas in +the galactic centre, likely required several jet episodes to occur, given +that each jet episode is collimated and directional. Like the CO(1-0) +molecular gas image, the GALEX NUV image also shows a clear +deficiency of star-formation in the last 200 Myr in the inner ∼ 6 kpc +region. Additionally, the SFR surface density is lower by a factor of +5 − 18 compared to the global Schmidt law of star-forming galaxies. +These results point to star-formation quenching taking place in the +central regions of NGC 2639. This makes NGC 2639 a rare case of +a radio-quiet AGN showing episodic jet activity and possible signa- +tures of negative AGN feedback. +ACKNOWLEDGEMENTS +We thank the anonymous referee for their insightful suggestions that +have improved this manuscript. We thank the staff of the uGMRT +that made these observations possible. uGMRT is run by the National +Centre for Radio Astrophysics of the Tata Institute of Fundamental +Research. PK, RK, JB, SS and SM acknowledge the support of the De- +partment of Atomic Energy, Government of India, under the project +12-R&D-TFR-5.02-0700. The National Radio Astronomy Observa- +tory is a facility of the National Science Foundation operated under +cooperative agreement by Associated Universities, Inc. +DATA AVAILABILITY +The data underlying this article will be shared on reasonable request +to the corresponding author. +REFERENCES +Alatalo K., et al., 2011, ApJ, 735, 88 +Alexander D. M., Hickox R. C., 2012, New Astron. Rev., 56, 93 +Antonucci R., 1993, ARA&A, 31, 473 +Berrier J. C., et al., 2013, ApJ, 769, 132 +Bolatto A. D., et al., 2017, ApJ, 846, 159 +Bower R. G., Benson A. J., Crain R. A., 2012, MNRAS, 422, 2816 +Braatz J. A., Wilson A. 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S., 1976, A&A, 53, 93 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–9 (2022) + diff --git a/i9AzT4oBgHgl3EQfpP2e/content/tmp_files/load_file.txt b/i9AzT4oBgHgl3EQfpP2e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..14a85ca00a92ba669355e4ec5d77b6545371bd59 --- /dev/null +++ b/i9AzT4oBgHgl3EQfpP2e/content/tmp_files/load_file.txt @@ -0,0 +1,1048 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf,len=1047 +page_content='MNRAS 000, 1–9 (2022) Preprint 5 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0 AGN Feedback Through Multiple Jet Cycles in the Seyfert Galaxy NGC 2639 Vaishnav V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Rao,1★P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Kharb,2 Rubinur K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=',3 Silpa S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=',2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Roy,4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Sebastian,5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Singh,6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Baghel,2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Manna,2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Ishwara-Chandra2 1Indian Institute of Technology Bombay, Powai, Mumbai 400076, India 2National Centre for Radio Astrophysics (NCRA) - Tata Institute of Fundamental Research (TIFR), S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Pune University Campus, Ganeshkhind, Pune 411007, India 3Institute of Theoretical Astrophysics, University of Oslo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='O box 1029 Blindern, 0315 OSLO, Norway 4Johns Hopkins University, Department of Physics & Astronomy, 3400 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Charles Street, Baltimore, MD 21218, USA 5Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB R3T 2N2, Canada 6Astronomy and Astrophysics Division, Physical Research Laboratory, Ahmedabad 380009, India 5 January 2023 ABSTRACT The Seyfert galaxy NGC 2639 is known to exhibit three episodes of AGN jet/lobe activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' We present here the upgraded Giant Metrewave Radio Telescope (uGMRT) 735 MHz image of NGC 2639 showing a fourth episode as witnessed by the discovery of ∼ 9 kpc radio lobes misaligned with the previously known ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 kpc, ∼ 360 parsec, and ∼ 3 parsec jet features detected through the Karl G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Jansky Very Large Array (VLA) and the Very Long Baseline Array (VLBA), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Using the spectral ageing software BRATS, we derive the ages of the ∼ 9 kpc, ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 kpc, and ∼ 360 parsec episodes to be, respectively, 34+4 −6 Myr, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='4 Myr, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 Myr, and conclude that minor mergers occurred 9-22 Myr apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' NGC 2639 shows a deficit of CO(1-0) molecular gas in its central ∼ 6 kpc region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The GALEX NUV image also shows a clear deficiency of recent star-formation in the same region, while the star formation rate (SFR) surface density in NGC 2639 is lower by a factor of 5 − 18 compared to the global Schmidt law of star-forming galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' This makes NGC 2639 a rare case of a radio-quiet AGN showing episodic jet activity and possible signatures of negative AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Key words: galaxies: Seyfert – galaxies: jets – radio continuum: galaxies – techniques: interferometric 1 INTRODUCTION Active galactic nuclei (AGN) are the high-luminosity, energetic centres of galaxies that are dominated by light emitted from mat- ter accreting onto a supermassive black hole (SMBH, MBH ∼ 106 − 109 M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Rees 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Padovani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' In a small fraction of AGN (∼ 10%), relativistic jets are launched from the SMBH up to hundreds of kpc to even Mpc scales (Readhead et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Orr & Browne 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Heckman & Best 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' AGNs have been histor- ically broadly classified into Seyfert galaxies and quasars, with the fundamental difference lying in their bolometric luminosity (Lbol) with Lbol ≤ 1012 L⊙ for Seyferts and higher for quasars (Schmidt & Green 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Soifer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Based on the presence or absence of broad bases to permitted emission lines in the optical spectra, Seyferts are further classified into types 1 and 2 (Khachikian & Weedman 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Hickox & Alexander 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' It is widely believed that different viewing angles of the central engine surrounded by an obscuring torus explain the two Seyfert classes, with type 1s being viewed nearly face-on, although there are several exceptions (An- tonucci 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Ho 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Netzer 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Seyfert galaxies are primarily classified as “radio-quiet” (RQ) AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Based on 6 GHz VLA obser- vations of SDSS quasi-stellar objects, Kellermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2016) have ★ E-mail: vaishnavrao@iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='in suggested RQAGN to be those having 21 ≤ log[L6(W Hz−1)] ≤ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Radio observations have revealed that most Seyferts possess sub- parsec or parsec-scale radio emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='g, de Bruyn & Wilson 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Ulvestad & Wilson 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Thean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Kharb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2021) and ≳ 40% display kpc-scale radio structures (KSRs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Gallimore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' AGN are believed to regulate galaxy growth by injecting energy into the surrounding gas which has the effect of either heating and/or expelling star-forming material (‘negative feedback’) or facilitating localized star-formation (‘positive feedback’) (Alexander & Hickox 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Fabian 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Morganti 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Harrison 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' While AGN ‘feedback’ is believed to be a fundamental process of galaxy for- mation, there are many outstanding questions from an observational point of view;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' unequivocal observational signatures are rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' AGN feedback has been suggested to come in ‘quasar mode’ and ‘mainte- nance/jet mode’ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=', Croton 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Bower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The former is associated with ‘radiative mode’ AGN like quasars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=', Faucher- Giguère & Quataert 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2018), while the latter is associated with AGN with radio jets that can transfer mechanical power (jets could heat, shock or entrain gas) and regulate star forma- tion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=', McNamara & Nulsen 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Mahony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Morganti 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Hardcastle & Croston 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' While highly collimated jets can- not be efficient agents of AGN “feedback”, presumably due to the smaller working surfaces at their advancing ends, relatively isotropic © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='01610v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='GA] 4 Jan 2023 2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 8h43m42s 39s 36s 33s 50°13\'00" 12\'30" 00" 11\'30" RA (J2000) Dec (J2000) 1 kpc 325MHz GMRT 735MHz GMRT 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5GHz VLA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='00 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='00 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' A radio-optical overlay of NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The optical image comes from SDSS r-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The dotted, white contours represent the 325 MHz uGMRT data with contour levels (1, 2, 4, 8) × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The solid white contours represent the 735 MHz uGMRT data with contour levels (1, 2, 4, 32) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The solid brown contours come from the VLA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 GHz image with contour levels (2, 4, 8, 64) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='03 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' impacts via changes in jet direction can be highly effective (King & Pounds 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Moreover, the contribution of jets towards the to- tal AGN energy budget has been suggested to be less than ∼ 10% (Cattaneo & Best 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Sanders (1984) have argued that individual Seyfert activity episodes typically have a shorter duration than the minimum sta- tistical lifetime of Seyfert activity in a particular galaxy (3 − 7 × 108 yr), which would imply that the nuclei evolve through at least a 100 recurring Seyfert episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' If it is assumed that Seyfert episodes in a particular galaxy are due to accretion onto the central black hole, the short lifetime of Seyfert events would require separate episodes to be caused by distinct accretion events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Recent studies from the SDSS, though, have suggested that accretion rate changes are com- mon within a given Seyfert duty cycle, producing a much more complex picture of accretion in Seyfert galaxies (Koulouridis 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Elitzur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Koulouridis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' So far, only a handful of Seyfert galaxies such as Mrk 6 (Kharb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2006) and NGC 2992 (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2016) have been known to exhibit multiple kpc-scale radio lobes, with Mrk 6 showing three jet episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Here and in other cases discussed in this paper, different jet “episodes” are defined by clear surface brightness discontinuities, with each jet episode being delineated by the presence of either terminal hotspot-like features or lobes with well-defined edges so that they could not be mistaken for sharp jet changes that may occur in a single epoch due to magneto- hydrodynamic (MHD) instabilities in the jet (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=', Leismann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Interestingly, Sebastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2020) found tentative evidence for multiple episodes in a majority (5/9) of their Seyfert galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' This fraction was higher than that observed in radio galaxies (∼ 10 − 15%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Jurlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2020) and was consistent with the theoretical expectation of Sanders (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Therefore, the rarity of known Seyfert galaxies with episodic activity may not be due to a true absence but rather due to the difficulty in their identification due to the low surface brightness of their lobes, small spatial extents, lack of collimation, and confusion with the radio emission from star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Hence, it is essential to design methods to identify various episodes of jetted emission in Seyfert galaxies to truly understand the life-cycle of jets in these systems and their impact on the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Multi-frequency, multi-scale/resolution observations are one such method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Although several AGN, including radio galaxies and Seyfert galaxies, have been shown to exhibit two episodes of jet activity, it is scarce to find sources with three episodes although see Lalakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' So far, none have been discovered with four episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Here we report the discovery of a Seyfert galaxy NGC 2639, which shows four AGN jet activity episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='1 NGC 2639 NGC 2639 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' UGC 4544) is a Seyfert 2 galaxy (Lacerda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2020, see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' At its redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='01113 (luminosity distance 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 Mpc), 1′′ corresponds to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='229 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' NGC 2639 hosts water vapour megamasers most likely in a cool dense nuclear disk (Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The megamaser in NGC 2639 is weak and of relatively low luminosity compared to other galaxies with megamasers (Braatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Berrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2013) have estimated the mass of the SMBH in NGC 2639 using the MBH − 𝜎 relation (Ferrarese & Mer- ritt 2000), which turns out to be MBH = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='43) × 108 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' A more accurate BH mass is not yet available from the water vapor megamaser data due to the non-detection of "satellite” lines, as de- scribed by Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The stellar mass of the host galaxy has been estimated to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='48×1011 M⊙ by Sweet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The Ed- dington ratio is given as, �𝑚 = �𝑀/ �𝑀𝐸𝑑𝑑 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7 𝛼𝑣 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='3 � 𝐿bol 𝐿Edd � (Ho 2009), where 𝛼𝑣 is the viscosity parameter , Lbol is the bolometric luminos- MNRAS 000, 1–9 (2022) Seyfert NGC 2639 3 ity and LEdd is the Eddington luminosity (= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='26 × 1038 � 𝑀𝐵𝐻 𝑀⊙ � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='86 × 1046 erg s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' We derive an accretion rate of �𝑚 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7 × 10−3 for NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Lbol was obtained using the 2-10 keV X-ray luminos- ity and LX (= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='1 × 1040 erg s−1 for NGC 2639) was estimated via the relation Lbol = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8 LX (Ho 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Assuming an optically thin radiatively inefficient accretion flow (RIAF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Narayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 1998) in NGC 2639, 𝛼𝑣2 is taken to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' For � 𝐿bol 𝐿Edd � = 10−6 − 10−4, �𝑚 ≃ 3 × 10−4 − 2 × 10−2, which lies within the regime of optically thin RIAFs, �𝑚 ≤ �𝑚crit ≃ 𝑎2𝑣 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='1 (Ho 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Using multi-resolution radio observations with the Karl G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Jansky Very Large Array (VLA) and the Very Long Baseline Array (VLBA), Sebastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2019) reported three pairs of radio jets/lobes in NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The three lobes are misaligned with each other without any apparent signatures of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Sebastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2019) con- cluded that these three lobes were representative of three distinct episodes of AGN activity and ruled out other scenarios, including multiple jets from independent AGN, slow jet precession, and jet bending due to pressure gradients within the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The jet extents on a single side of the three episodes were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8 kpc, 180 pc, and 3 pc as revealed by the VLA image at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 GHz, historical VLA image at 5 GHz, and VLBA image at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='3 GHz, respectively (see Figure 2 and Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Following the criterion of Kellermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2016), NGC 2639 is a radio-quiet AGN having log[L6(W Hz−1)] = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='15, obtained from its 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 GHz VLA lobes using a spectral index of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' In this paper, we present new uGMRT images of NGC 2639 which reveal an additional set of radio lobes, not previously detected at GHz frequencies (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The paper is divided as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' In Section 2, we present the radio data analysis, followed by the spectral ageing analysis in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The results and discussion follow in Sections 4 and 5, respectively, while the conclusions follow in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' In this paper, we have adopted the following cosmological parameters: 𝐻0 = 73 km s−1 Mpc−1, Ωmat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='27, Ωvac = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Throughout the paper, spectral index 𝛼 is defined such that flux density 𝑆𝜈 ∝ 𝜈𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2 RADIO OBSERVATIONS AND DATA ANALYSIS NGC 2639 was observed with the uGMRT (Project code: 39_090) at 735 MHz (band 4) on November 15, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The observing band consisted of a single spectral window, ranging from 550 MHz to 950 MHz, across 4096 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The total on-source time was ≈36 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 3C 147 was used as the amplitude calibrator and 0834+555 as the phase calibrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Initial data editing and calibration were car- ried out using the CAPTURE continuum imaging pipeline for uGMRT (Kale & Ishwara-Chandra 2021) on CASA (version 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' McMullin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The multi-term multi-frequency synthesis (MT-MFS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' see Rau & Cornwell 2011, for more details) algorithm with two Taylor terms was used while imaging in CASA to account for wide-band related errors while deconvolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Four rounds of phase-only self- calibration followed by four rounds of A&P (amplitude and phase) self-calibration were performed before the final image of NGC 2639 was created using the tclean task in CASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' tclean is the radio inter- ferometric image reconstruction task that contains standard “clean” based algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=', Högbom 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Clark 1980) along with algo- rithms for multi-scale and wideband image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The final rms noise in the resulting image was ∼ 90 𝜇Jy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Calibrated data of NGC 2639 used by Sebastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2019) from the VLA B-array at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 GHz, were was also available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The data was imaged using the tclean task in CASA using similar steps as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Images obtained from archival VLA data at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 GHz from 1985 and uGMRT data at 325 MHz were used along with the above data sets for a spectral-ageing analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' In addition to this, VLA images at 5 GHz from 1998 and VLBA images at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='3 GHz from 2011, which were analyzed by Sebastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2019), were included in our analysis as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' These figures are included as insets in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Table 1 includes a summary of the datasets used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 3 SPECTRAL AGEING ANALYSIS USING BRATS We used the Broadband Radio Analysis ToolS (BRATS) Software (Harwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2013, 2015) to carry out a spectral ageing analysis of the different jet episodes in NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' We estimated the ‘mini- mum energy’ magnetic field (𝐵min) in the radio lobes assuming the equipartition of magnetic field and particle energy densities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=', Pa- cholczyk 1970) and used this as the magnetic field input in BRATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' We used the relations provided in O’Dea & Owen (1987) to calculate 𝐵min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The volume filling factor, 𝜙, and the ratio of energy density of ions to electrons, 𝜂, was assumed to be unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The upper and lower radio-band frequency cutoffs, 𝜈𝑢 and 𝜈𝑙 were taken to be 10 MHz and 100 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The lobes were assumed to be cuboidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The input parameters, 𝐵min, and other estimates are noted in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='1 uGMRT North-East South-West Lobes To perform a spectral age analysis using BRATS, a minimum of three images of same size, resolution, and beam size at different frequencies are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' From the calibrated uGMRT band-4 data of NGC 2639, two subband images at central frequencies 643 MHz and 810 MHz were obtained using tclean with similar parameters as section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' These two images were smoothed and regridded using imsmooth and imregrid on CASA to match the resolution, image size, and beam size of the lower resolution 325 MHz archival uGMRT image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The final common restoring beam used was 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='1′′ × 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8′′ at PA = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='62◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' In doing so, it was assumed that the lower resolution 325 MHz uGMRT image contained the unresolved lobes which were seen at 735 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The Jaffe-Perola (JP) model was used for fitting the synchrotron spectrum of the three radio images on BRATS (see Harwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2013, 2015, for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' A 5𝜎 detection threshold was used with only the source regions selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The injection index was chosen as −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 as this was the approximate spectral index of the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' In doing so, it was assumed that electrons are being accelerated near the center as opposed to the progressing edges of the lobes as in FRII radio galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=', Mahatma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' This was because no hotspots were observed in any of the radio images and the source resembles an FRI-type or Seyfert-like jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The magnetic field (bfield) was set to the equipartition value of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6 × 10−6 G and the spectral age map was obtained using the fitjpmodel task on BRATS with levels=5 and Myears=60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 VLA North-South Lobes From the calibrated 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 GHz full-band VLA data of NGC 2639, two subband images at central frequencies 5 GHz and 6 GHz were obtained using tclean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' These two images were smoothed and regrid- ded using imsmooth and imregrid on CASA to match the resolution, image size, and beam size of the lower resolution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 GHz archival VLA image, in which only the southern lobe was visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The fi- nal common restoring beam used was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='45′′ × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='12′′ at PA 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='89◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' With the core and southern lobe regions selected and a 5𝜎 detection threshold, the synchrotron spectrum of the radio images were fitted with the JP model on BRATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The injection index was again chosen MNRAS 000, 1–9 (2022) 4 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The four AGN jet episodes of NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (Left) 735 MHz uGMRT total intensity image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The ∼ 9 kpc radio lobes are seen in this image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Contour levels: (−2, −1, 1, 2, 4, 8, 16, 32, 64, 128, 256) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Beam at the bottom left corner is of size: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='48′′ × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0′′ at PA=54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (Top) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 GHz VLA total intensity image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Contour levels: (−2, −1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='03 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 kpc north-south radio jets are seen here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Beam size: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='02′′ × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='89′′ at PA=−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (Right) 5 GHz VLA radio image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Contour levels: (−2, −1, 1, 2, 4, 8, 16, 32, 64, 128) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='164 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The ∼ 360 parsec east-west lobes are seen in this image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Beam size: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='43′′ × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='30′′ at PA=−85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='4◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (Bottom) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='3 GHz VLBA image showing a ∼ 3 parsec jet at PA = 130◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Contour levels: (−2, −1, 1, 2, 4, 8, 16, 32, 64) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='239 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Beam size: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7 mas×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 mas at PA=−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='9◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Details of the different datasets used for analysis in this paper Telescope Project Code Frequency Array Beam size rms noise (𝜇Jy beam−1) ∗uGMRT 39_090 735 MHz 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='48′′ × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0′′, PA 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6◦ 90 ∗uGMRT 22_002 325 MHz 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='1′′ × 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8′′, PA 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='62◦ 128 ∗VLA 17B-074 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 GHz B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='02′′ × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='89′′, PA −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8◦ 9 VLA GL022 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0 GHz A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='43′′ × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='30′′, PA −85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='4◦ 54 ∗VLA AW126 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 GHz A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='45′′ × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='12′′, PA 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='89◦ 82 VLBA BC196J 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='3 GHz 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7 mas × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 mas, PA −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='9◦ 75 ∗ indicates the datasets that have been used for the BRATS spectral ageing analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' to be −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 and the magnetic field was taken as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 × 10−5 G from the equipartition estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The spectral age map was obtained from the fitjpmodel task with levels=5 and Myears=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 4 RESULTS Figure 1 shows the total radio intensity image contours at 325 MHZ, 735 MHz, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 GHZ superimposed on the SDSS r-band color map of NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Figure 2 shows the total intensity image from uGMRT, as well as the total intensity images from VLA and VLBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The new northwest-southeast radio lobes, which were previously not detected in GHz frequency observations with the VLA, are clearly seen in Figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The linear extents of each of these lobes are ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The position angle (PA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' measured from north through east with north being at 0◦) of the host galaxy disc is 136◦ whereas the PA of the northwest-southeast lobes is 34◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The PA of the VLA north- south lobes is −174◦ and that of the east-west lobes is 106◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' uGMRT images at 325 MHz do not resolve these lobes clearly but detect additional radio emission from the host galaxy itself (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='1 Episodic Activity and AGN Jet Duty Cycle The uGMRT, VLA, and VLBA data show that NGC 2639 is a can- didate for an AGN with 4 jet episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' We note that it is possible that the VLBA jet is feeding the east-west VLA lobes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=', Kharb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2010), which could reduce the number of episodes to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' However, in view of the ∼ 30◦ PA offset between the VLBA jet and the VLA east-west lobes, we will continue to refer to four jet episodes in NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Using these multi-frequency arcsec-scale radio data, MNRAS 000, 1–9 (2022) Seyfert NGC 2639 5 8h43m40s 39s 38s 37s 36s 50°12\'45" 30" 15" 00" RA (J2000) Dec (J2000) JP Model Injection Index: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 B-field: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='60 × 10 6 G 1 kpc 643MHz subband GMRT 10 20 30 40 50 60 Spectral Age (Myr) 8h43m40s 39s 38s 37s 36s 50°12\'45" 30" 15" 00" RA (J2000) Dec (J2000) JP Model Injection Index: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 B-field: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='60 × 10 6 G 1 kpc 643MHz subband GMRT 10 20 30 40 50 60 Positive Error (Myr) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (Left) Spectral age map of the uGMRT ∼ 9 kpc lobes obtained using the smoothed and regridded images at 325, 643, and 810 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (Right) The map of the positive error in estimated spectral age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The mean spectral age of the jet is 34+4 −6 Myr (Note: The purple patches at the top and bottom of the spectral age map along with the yellow sliver on the right of the map were excluded from calculating mean due to poor fitting and high errors as can be seen in the right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Contour levels are of the 643 MHz uGMRT subband image (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='1) at: (1, 2, 4, 8, 16, 32, 64) × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 8h43m38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6s 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='4s 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2s 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0s 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8s 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6s 50°12\'24" 21" 18" 15" RA (J2000) Dec (J2000) JP Model Injection Index: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 B-field: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='21 × 10 5 G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 kpc 5GHz subband VLA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0 Spectral Age (Myr) 8h43m38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6s 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='4s 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2s 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0s 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8s 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6s 50°12\'24" 21" 18" 15" RA (J2000) Dec (J2000) JP Model Injection Index: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 B-field: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='21 × 10 5 G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 kpc 5GHz subband VLA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0 Positive Error (Myr) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (Left) Spectral age map of the VLA core and the ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8 kpc south lobe obtained using the smoothed and regridded images at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (Right) The map of the positive error in estimated spectral age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The mean spectral age of the southern lobe is 12+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='4 Myr and the mean spectral age of the core is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Contour levels are of the 5 GHz VLA subband image (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2) at: (1, 3, 9, 27, 81) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='036 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' we have carried out a spectral ageing analysis using BRATS, as de- scribed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Figures 3 and 4 show the spectral age maps of the uGMRT and VLA lobes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The mean spectral age of the northeast-southwest uGMRT lobes is 34+4 −6 Myr, that of the southern VLA lobe is 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='4 Myr, and that of the core is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The ages of the southern lobe and core obtained are comparable to the electron lifetime estimates obtained by Sebastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2019) of 12- 16 Myr and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8 Myr respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' These results indicate that the jets were launched 9-22 Myr apart with the 9 kpc, northwest-southeast, uGMRT jets being launched first followed by the ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 kpc, north- south VLA jets, and the 360 parsec east-west VLA jets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Defining the AGN jet duty cycle as 𝜖 = 𝑡on/(𝑡on + 𝑡off) (see Clarke & Burns 1991), we use the ages of the VLA east-west lobes and north-south lobes to obtain 𝜖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Similarly, using the ages of the VLA north-south lobes and the uGMRT lobes, we obtain 𝜖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The AGN jet duty cycle in NGC 2639 is therefore ∼ 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' These data however, cannot directly address to other AGN activity episodes that might not have produced radio jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' For instance, these data cannot rule out episodic “wind” activity that may be unrelated to the radio jets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=', King & Pounds 2015), leaving an overall uncertainty in the true AGN duty cycle in NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 Looking for the Jet-Gas Connection Using the kpc-scale measurements of molecular gas and stellar mass surface densities taken from the EDGE (Bolatto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2017) - Calar Alto Legacy Integral Field Area (CALIFA Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2012) sur- MNRAS 000, 1–9 (2022) 6 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Equipartition estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Lobes were assumed to be cuboidal with volume filling factor, 𝜙 = 1 and the ratio of energy density of ions to electrons, 𝜂 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The constant 𝑐12 which is a function of spectral index and radio-band frequency cutoffs has been obtained from Pacholczyk (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Lobe Frequency Length Width Flux density 𝛼 𝑐12 𝐿rad 𝐵min kpc kpc mJy erg s−1 G uGMRT (both lobes) 735 MHz 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='1 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='3 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8 × 1040 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6 × 10−6 VLA-south 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6 × 107 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8 × 1038 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='21 × 10−5 8h43m40s 38s 36s 50°12\'45" 30" 15" 00" RA (J2000) Dec (J2000) 1 kpc 735MHz GMRT 0 1 2 3 4 5 Jy/bm km/s 8h43m40s 38s 36s 50°12\'45" 30" 15" 00" RA (J2000) Dec (J2000) 1 kpc 735MHz GMRT 10 20 30 40 50 60 km/s Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (Left) Moment-0 image of the CO(1-0) molecular gas emission line from the CARMA EDGE survey, overlaid with the 735 MHz uGMRT radio contours at (1, 2, 4, 32) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' This image represents the integrated CO intensity and shows a deficiency of CO(1-0) molecular gas in the central ∼6 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (Right) The Moment-2 image with the same contour levels, representing velocity dispersion of the CO(1-0) molecular gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Higher velocity dispersion values are observed around the jet edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 8h43m40s 38s 36s 50°12\'45" 30" 15" 00" RA (J2000) Dec (J2000) 1 kpc 735MHz GMRT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='08 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Near-UV image of NGC 2639 from GALEX overlaid with the 735 MHz uGMRT radio contours at (1, 2, 4, 32) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='6 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The central ∼6 kpc shows a deficiency of NUV emission indicating quenching of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' vey1, Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2021) have shown that the gas fractions of central AGN regions in their sample galaxies, including NGC 2639, are typ- 1 https://mmwave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='edu/edgedata/ ically a factor of ∼ 2 lower than in star-forming regions, suggesting that the AGN has partially depleted the central molecular gas reser- voir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The depletion of molecular gas from the central AGN region was noted to be more pronounced for NGC 2639, indicating increased outflow rates compared to the other AGN-host galaxies in their sam- MNRAS 000, 1–9 (2022) Seyfert NGC 2639 7 ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The left panel of Figure 5 shows the Moment-0 image of the CO(1-0) molecular gas emission line from the CARMA EDGE sur- vey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' This image represents the integrated CO intensity and clearly shows the deficit of CO(1-0) molecular gas in the central ∼ 6 kpc region of NGC 2639;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' the integrated intensity is 5-7 times higher in the ring than the central regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The Moment-1 map of NGC 2639 suggests rotating molecular gas that appears to be largely regular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' clear signatures of turbulence are not directly observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The Moment-2 map however, reveals slightly higher velocity-dispersion values (of the order of 100 km s−1) around the uGMRT jet edges (see right panel of Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' This could suggest that the jet does impact the molecular gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' If the CO(1-0) molecular gas ring is a result of “push-back” from the jet in NGC 2639, we can estimate the 𝑃𝑉 (pressure times volume) amount of work done on the molecular gas by the jet to create a cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Using the typical temperature (𝑇) and density (𝑛) of the molecular medium of the galactic ISM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=', 𝑇 ≃ 20 K, and 𝑛 > 103 cm−3 (see Brinks 1990), and assuming it to be similar in NGC 2639, we can estimate the CO gas pressure as 𝑃 = 𝑛𝑘𝐵𝑇, where 𝑘𝐵 is the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' For the CO gas ring radius of 3 kpc, the volume of the disk-like cavity is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='25 × 1066 cm3 and the 𝑃𝑉 work done is > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='44 × 1054 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' We can also estimate the jet mechanical powers using the relations derived for low luminosity AGN by Merloni & Heinz (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Using only the lobe flux densities at 5 GHz, we find that the jet power for the east-west VLA lobes is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7 × 1042 erg s−1 and the time-averaged power (for a spectral age of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8 Myr) is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8×1056 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Similarly, for the north-south VLA lobes, the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 GHz jet power is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='3 × 1042 erg s−1, and its time-averaged power (over 12 Myr) is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='9 × 1056 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Finally, using an average lobe spectral index of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='3, the jet power for the north-east - south-west uGMRT lobes is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7 × 1042 erg s−1, and its time-averaged power (over 34 Myr) is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='1×1057 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Therefore, only ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5% of the east-west jet power is sufficient to push back the CO gas in NGC 2639;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' these numbers are ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7% for the north-south jets and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='06% for the north-east - south-west jets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='3 Indications of Star-formation Quenching The GALEX data on NGC 2639 shows a clear deficiency of far- and near-UV (FUV and NUV) emission from the central ∼ 6 kpc of the galaxy (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The NUV band directly traces stars formed over the last 200 Myr and therefore probes recent star formation (Kennicutt & Evans 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The central void in the GALEX NUV image, therefore, is consistent with the suggestion of star-formation quenching in the central few kpc in NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' We attempt to quantify this further ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The global Schmidt law for star-forming galaxies has been given by Kennicutt (1998) as: ΣSFR = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7) × 10−4 � Σgas 1 M⊙ pc−2 �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='15 M⊙ yr−1 kpc−2 (1) where the SFR surface density, ΣSFR, can be derived from gas sur- face density, Σgas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' For NGC 2639, using Σgas ≡ ΣH2 = 21 M⊙ pc−2 for a region of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8 kpc (Raluy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 1998), ΣSFR should be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0177 M⊙ yr−1 kpc−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The SFR surface density can also be com- puted from the SFR estimates using the following expression: ΣSFR = SFR 𝜋a2 � d 206265 �2 (2) where the parameter 𝑎 corresponds to the semi-major axis of the telescope aperture in arcsec and 𝑑 is the distance to the galaxy in Mpc (Catalán-Torrecilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Several estimates for SFR have been derived for NGC 2639 in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Table 3 provides the estimates of ΣSFR obtained using different SFR tracers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The telescope details for individual tracers are also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' These estimates of ΣSFR are suggesting that star-formation is quenched in NGC 2639 compared to the global Schmidt law for star-forming galaxies by a factor of 5 − 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' This is fully consistent with the GALEX NUV image of NGC 2639, showing a deficit in recent star-formation in the central ∼ 6 kpc region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 5 DISCUSSION Sebastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2019) have argued that the multiple radio lobes seen in NGC 2639 are due to minor mergers that did not disrupt the morphology of the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' We also find that the line of sight velocity map of the host galaxy from the CALIFA survey shows that the stars of NGC 2639 are relatively undisturbed and are more or less in uniform motion (de Amorim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The misaligned jets are the result of new accretion disks formed from mergers, with jet directions conserving the angular momentum of the inflowing gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' This scenario is consistent with NGC 2639 having a large bulge component surrounding the nucleus (Cox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2007) as gravitational forces and torques that result from mergers disrupt the orbital path of stars causing randomised bulge orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Thus, if the minor merger scenario were true, the spectral age results of the multiple lobes indicate that minor mergers occurred every 9 − 22 Myr apart in the last ∼ 30 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The expected minor merger rate for a galaxy like NGC 2639 (red- shift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='01113 and stellar mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='48 × 1011 M⊙) is ∼ 13 Myr following the work of Conselice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2022), who used observa- tional data from the REFINE survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' We used their minor mergers best fit line (with stellar mass ratios of 1:10) to obtain this estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The estimate of ∼ 10 Myr also matches the estimates obtained via theoretical studies as well as galaxy-merger simulations (Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Capelo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' It would therefore be fair to conclude that at least three minor mergers have taken place in the lifetime of NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Each of these mergers may have resulted in the formation of a new accretion disk with no memory of the previous accretion disk direction, primarily driven by the angular momentum of the infalling material itself (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=', Kharb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Accretion through these disks would have resulted in the several differently-oriented jet episodes that are observed in NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' As noted in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2, each of these jet episodes have sufficient mechanical power to dis- place the CO molecular gas from the central few kpc of the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' However, due to the directionality of the collimated jets, it takes multiple differently-oriented jets to create the CO gas ring seen in NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The early-type galaxy and LINER, NGC 1266, shows the presence of a CO molecular outflow, no signatures of galaxy interactions, and a possible radio jet at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='4 GHz (Alatalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' There is also a centrally concentrated molecular component which is different from the case of NGC 2639, where instead, a central deficiency is ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Alatalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2011) suggest that (the sole) jet in NGC 1266 is sufficient to drive the molecular outflow using only 2% of its total power at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' We note, however, that multi-resolution, multi- frequency radio observations are required to truly rule out the exis- tence of multiple jet episodes in NGC 1266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Nesvadba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2021) detected a CO(1-2) molecular gas ring through ALMA observations in the nearby spiral galaxy J2345−0449 with large kpc-scale radio jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Interestingly, the inner radius of the CO gas ring corresponds to MNRAS 000, 1–9 (2022) 8 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Estimates of star formation rate in NGC 2639 using different tracers SFR Telescope SFR indicator Aperture ΣSFR Reference (M⊙ yr−1) (′′) (M⊙ yr−1 kpc−2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='92 Nickel 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0 m telescope at Lick Observatory H𝛼 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='00099 (1) 1 Spitzer Space Telescope IR 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0036 (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='57 Calar Alto 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 m telescope H𝛼 36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='0026 (3) References: (1) Theios et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2016) using the relation, SFR (M⊙ yr−1) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='37 × 10−42 LH𝛼 (erg s−1), where LH𝛼 = 1041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='23 (erg s−1) for NGC 2639 (2) Sebastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2019) using the CLUMPYDREAM code (3) Catalán-Torrecilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2015) using the H𝛼 line luminosity from the CALIFA survey 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='2 kpc, very similar to what is observed in NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' More- over, they find that the molecular gas outflow in J2345−0449 has a kinetic energy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='3 × 1057 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Again, only a small fraction of the jet kinetic power in J2345−0449 (and as it happens in NGC 2639) can suffice to drive molecular gas outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' It is worth noting that the radio source in J2345−0449 is also a restarted double-double radio galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The absence of CO(1-0) gas in the inner 6 kpc region of NGC 2639 could indicate the presence of “negative AGN feedback” by the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The relatively small values of ΣSFR observed in NGC 2639 com- pared to the Schmidt law for star-forming galaxies suggest that star- formation quenching is taking place in NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Simulations of Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Meenakshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' (2022) have shown that the jet-ISM coupling is sensitive to the relative orientation of the jet w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='t the gas disk, as well as the power and age of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' It is stronger for the jets, which are oriented at ≥ 45◦ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='t the gas disk, young (≤ 2 Myr), and highly powerful (≥ 1045 erg s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' In NGC 2639, a single jet episode may not meet all the above criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' However, three jet episodes together can result in an efficient coupling with the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' 6 CONCLUSIONS The Seyfert galaxy NGC 2639 exhibits four episodes of AGN jet ac- tivity as evidenced by 735 MHz, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 GHz, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='3 GHz frequency observations via the uGMRT, VLA and the VLBA telescopes, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Using the spectral ageing software BRATS, we derive the ages of the three pairs of lobes to be respectively, 34+4 −6 Myr, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='4 Myr, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='5 Myr, with the uGMRT lobes being the oldest (we did not derive an age for the VLBA jet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Using the “on” and “off” times of these jets/lobes, the AGN jet duty cycle in NGC 2639 is ∼ 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' NGC 2639 also shows a deficiency of molecular gas in its central ∼ 6 kpc region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Less than 1% of the jet mechanical power for each of the jet episodes taken individually, is sufficient to move the molecular gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' However, the creation of a ring in the molecular gas in the galactic centre, likely required several jet episodes to occur, given that each jet episode is collimated and directional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Like the CO(1-0) molecular gas image, the GALEX NUV image also shows a clear deficiency of star-formation in the last 200 Myr in the inner ∼ 6 kpc region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' Additionally, the SFR surface density is lower by a factor of 5 − 18 compared to the global Schmidt law of star-forming galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' These results point to star-formation quenching taking place in the central regions of NGC 2639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' This makes NGC 2639 a rare case of a radio-quiet AGN showing episodic jet activity and possible signa- tures of negative AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the anonymous referee for their insightful suggestions that have improved this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' We thank the staff of the uGMRT that made these observations possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' uGMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' PK, RK, JB, SS and SM acknowledge the support of the De- partment of Atomic Energy, Government of India, under the project 12-R&D-TFR-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content='02-0700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' The National Radio Astronomy Observa- tory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article will be shared on reasonable request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' REFERENCES Alatalo K.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=', 1976, A&A, 53, 93 This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} +page_content=' MNRAS 000, 1–9 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AzT4oBgHgl3EQfpP2e/content/2301.01610v1.pdf'} diff --git a/i9E3T4oBgHgl3EQfJAlP/vector_store/index.faiss b/i9E3T4oBgHgl3EQfJAlP/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..5fa3fe5930581006b37afbe0f6b12954fde232ea --- /dev/null +++ b/i9E3T4oBgHgl3EQfJAlP/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:995d8991bc07e5194316231fd03d0e1242674d3f7d5d4855e870eaf51c0f1a1b +size 4980781 diff --git a/i9E4T4oBgHgl3EQfsQ0k/content/2301.05214v1.pdf 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Quadrature formulas (QFs) based on radial basis functions (RBFs) have become an essential tool for multivariate +numerical integration of scattered data. Although numerous works have been published on RBF-QFs, their +stability theory can still be considered as underdeveloped. Here, we strive to pave the way towards a more +mature stability theory for global and function-independent RBF-QFs. +In particular, we prove stability of +these for compactly supported RBFs under certain conditions on the shape parameter and the data points. +As an alternative to changing the shape parameter, we demonstrate how the least-squares approach can be +used to construct stable RBF-QFs by allowing the number of data points used for numerical integration to be +larger than the number of centers used to generate the RBF approximation space. Moreover, it is shown that +asymptotic stability of many global RBF-QFs is independent of polynomial terms, which are often included in +RBF approximations. While our findings provide some novel conditions for stability of global RBF-QFs, the +present work also demonstrates that there are still many gaps to fill in future investigations. +Key words. Numerical integration, radial basis functions, stability, cardinal functions, discrete orthogonal polynomials +AMS subject classifications (2020). 65D30, 65D32, 65D05, 42C05 +1. Introduction. Numerical integration is an omnipresent task in mathematics and myriad appli- +cations. While these are too numerous to list fully, prominent examples include numerical differential +equations [46, 76, 1], machine learning [68], finance [34], and biology [62]. In many cases, the problem +can be formulated as follows. Let Ω ⊂ RD be a bounded domain with positive volume, |Ω| > 0. Given +N distinct data pairs {(xn, fn)}N +n=1 ⊂ Ω×R with f : Ω → R and fn := f(xn), the aim is to approximate +the weighted integral +I[f] := +� +Ω +f(x)ω(x) dx +by an N-point QF. That is, by a weighted finite sum over the given function of the form +CN[f] = +N +� +n=1 +wnf(xn). +In higher dimensions, CN is sometimes referred to as an N-point cubature formula. The distinct points +{xn}N +n=1 are called data points and the {wn}N +n=1 are referred to as quadrature weights. Many QFs are +derived based on the idea to approximate the (unknown) function f and then exactly integrate this +approximation [43, 88, 25, 18, 57, 19, 58, 21, 9, 91]. Arguably, most of the existing QFs have been +derived from being exact for polynomials up to a certain degree. See [63, 69, 20, 70, 19, 93], in addition +to the above references. +That said, in recent years, QFs based on the exact integration of RBFs have received a growing +amount of interest [87, 85, 84, 75, 2, 32, 78, 80, 95, 79, 86]. The increased use of RBFs for numerical +∗January 31, 2023 +Corresponding author: Jan Glaubitz (Jan.Glaubitz@Dartmouth.edu, orcid.org/0000-0002-3434-5563) +Disclaimer: The views expressed in this academic research paper are those of the authors and do not reflect the official +policy or position of the United States Government or Department of Defense. In accordance with the Air Force Instruction +51-303, it is not copyrighted, but is the property of the United States government. +†Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA +‡Air Force Institute of Technology, Wright–Patterson Air Force Base, OH 45433, USA. +1 + +2 +J. GLAUBITZ AND J. A. REEGER +integration and numerical differential equations [55, 53, 26, 54, 60, 51, 83, 31, 28, 39, 40] seems to be +only logical, considering their story of success in the last few decades. In fact, since their introduction +in Hardy’s work on cartography from 1971 (see [45]), RBFs have become a powerful tool in numerical +analysis, including multivariate interpolation and approximation theory [12, 13, 98, 27, 52, 30]. It should +also be mentioned that RBF-QF can be connected to (statistical) Bayesian quadrature [72, 67, 10, 56]. +Finally, recent literature in the quadrature area [78, 80, 79, 77] has focused on ’local’ RBF-FD-type +implementations to reduce computational costs for large node numbers. While an extension to such local +approaches would be of interest, we restrict ourselves to global RBF methods in this work. That said, to +reduce the cost of constructing and integrating a global interpolant, a piecewise RBF interpolant could +be considered and integrated in a manner similar to the construction of Newton–Cotes formulas. Some +of our results would easily carry over to this setting, which might be seen as an extreme version (no +overlap of nonzero measure) of RBF partition of unity methods [3, 97, 27] or the overlapped RBF-FD +methods [82]. +Even though RBF-QFs have been proposed and applied in numerous works, their stability theory +can still be considered as under-developed, especially compared to more traditional—e. g polynomial +based—methods. Stability of RBF-QFs was broached, for instance, in [87, 85, 75]. Further, stability +of RBF-QF was discussed in [32] for integration on certain manifolds. However, to the best of our +knowledge, an exhaustive stability theory for RBF-QFs is still missing in the literature. In particular, +theoretical results providing clear conditions under which stability of RBF-QFs is ensured are rarely +encountered, even for global RBF methods. +The present work strives to fill this gap in the RBF literature partially. This is done by providing a +detailed theoretical and numerical investigation on stability of global RBF-QFs1 for different families of +kernels, including compactly supported and Gaussian RBFs as well as polyharmonic splines (PHS). Our +analysis resembles classic stability theory for quadratures exact for polynomial spaces. In contrast to +some existing works (see [15] and references therein), we consider RBF approximations with function- +independent shape parameters to obtain quadrature formulas that do not have to be recomputed when +another function is considered. +In particular, we report on the following findings. (1) We provide a sufficient condition for compactly +supported RBFs to yield a provable stable RBF-QF (see Theorem 4.1 in §4). The result is independent +of the degree of the polynomial term that is included in the global RBF interpolant and assumes the +data points to come from an equidistributed (space-filling) sequence. (2) We demonstrate how the idea +of least squares can be employed to construct provable stable RBF-QFs. (3) Asymptotic stability of pure +RBF-QFs is connected to asymptotic stability of the same RBF-QF but augmented with polynomials +of a fixed arbitrary degree. Essentially, we can show that for a sufficiently large number of data points, +stability of RBF-QFs is independent of the presence of polynomials in the RBF interpolant. +The rest of this work is organized as follows. We collect some preliminaries on RBF interpolants +and QFs in §2. In §3, a few initial comments on stability of (RBF-)QFs are offered. Next, §4 contains +our main theoretical result regarding stability of RBF-QFs based on compactly supported kernels. §5 +demonstrates how the concept of least squares can be used to construct provable stable RBF-QFs. +Furthermore, it is proven in §6 that, under certain assumptions, asymptotic stability of RBF-QFs is +independent of the polynomial terms included in the RBF interpolant. Numerical tests in §7 accompany +the previous theoretical findings. Finally, concluding thoughts are offered in §8. +2. Preliminaries. We collect some preliminaries on RBF interpolants (§2.1) and RBF-QFs (§2.2). +1Henceforth, we will refer to these as “RBF-QFs”. + +TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS +3 +2.1. Radial basis function interpolation. RBFs are often considered a powerful tool in numerical +analysis, including multivariate interpolation and approximation theory [12, 13, 98, 27, 52, 30]. We are +especially interested in RBF interpolants. Let f : RD ⊃ Ω → R be a scalar valued function. Given a set +of distinct data points (sometimes also referred to as centers), the RBF interpolant of f is of the form +(2.1) +(sN,df)(x) = +N +� +n=1 +αnϕ(εn∥x − xn∥2) + +K +� +k=1 +βkpk(x). +Here, ϕ : R+ +0 → R is the RBF (also called kernel), {pk}K +k=1 is a basis of the space of algebraic polynomials +up to degree d, Pd(Ω), and the εn’s are nonnegative shape parameters.2 The RBF interpolant (2.1) is +uniquely determined by the conditions +(sN,df)(xn) = f(xn), +n = 1, . . . , N, +(2.2) +N +� +n=1 +αnpk(xn) = 0, +k = 1, . . . , K. +(2.3) +Note that (2.2) and (2.3) can be reformulated as a linear system for the coefficient vectors α = +[α1, . . . , αN]T and β = [β1, . . . , βK]T . This linear system is given by +(2.4) +� +Φ +P +P T +0 +� � +α +β +� += +� +f +0 +� +, +where f = [f(x1), . . . , f(xN)]T as well as +(2.5) +Φ = + + +ϕ(ε1∥x1 − x1∥2) +. . . +ϕ(εN∥x1 − xN∥2) +... +... +ϕ(ε1∥xN − x1∥2) +. . . +ϕ(εN∥xN − xN∥2) + + , P = + + +p1(x1) +. . . +pK(x1) +... +... +p1(xN) +. . . +pK(xN) + + . +For a constant shape parameter ε1 = · · · = εN, (2.4) is ensured to have a unique solution—corresponding +to existence and uniqueness of the RBF interpolant—if the kernel ϕ is conditionally positive definite of +order d and the set of data points is Pd(Ω)-unisolvent. See, for instance, [27, Chapter 7] and [35, Chapter +3.1] or references therein. In this work, we shall focus on the popular choices of RBFs listed in Table +1. A more complete list of RBFs and their properties can be found in the monographs [13, 98, 27, 30] +and references therein. +The set of all RBF interpolants (2.1) forms an N-dimensional linear space, denote by SN,d. This +space is spanned by the cardinal functions +(2.6) +cm(x) = +N +� +n=1 +α(m) +n +ϕ(εn∥x − xn∥2) + +K +� +k=1 +β(m) +k +pk(x), +m = 1, . . . , N, +which are uniquely determined by the cardinal property +(2.7) +cm(xn) = δmn := +� +1 +if m = n, +0 +otherwise, +m, n = 1, . . . , N, +2For polyharmonic splines, it is common practice to not include a shape parameter in (2.1). For simplicity, we still use +(2.1) and set εn = 1, n = 1, . . . , n, in this case. + +4 +J. GLAUBITZ AND J. A. REEGER +RBF +ϕ(r) +parameter +order +Gaussian +exp(−r2) +0 +Wendland’s +ϕD,k(r), see [96] +D, k ∈ N0 +0 +Polyharmonic splines +r2k−1 +k ∈ N +k +r2k log r +k ∈ N +k + 1 +Table 1: Some popular RBFs. The “order” k of an RBF refers to the RBF being conditionally positive of order +k. +and condition (2.3). They provide us with the following representation of the RBF interpolant (2.1): +(sN,df)(x) = +N +� +n=1 +f(xn)cn(x) +This representation is convenient to subsequently derive quadrature weights based on RBFs that are +independent of the function f. +2.2. Quadrature formulas based on radial basis functions. A fundamental idea behind many QFs +is to first approximate the (unknown) function f : Ω → R based on the given data pairs {xn, fn}N +n=1 ⊂ +Ω×R and to exactly integrate this approximation. In the case of RBF-QFs this approximation is chosen +as the RBF interpolant (2.1). Hence, the corresponding RBF-QF is defined as +(2.8) +CN[f] := I[sN,df] = +� +Ω +(sN,df)(x)ω(x) dx. +When formulated w. r. t. the cardinal functions cn we get +(2.9) +CN[f] = +N +� +n=1 +wnf(xn) +with +wn = I[cn]. +That is, the RBF quadrature weights w = [w1, . . . , wN]T are given by the moments corresponding to +the cardinal functions. This formulation is often preferred over (2.8) since the weights w do not have +to be recomputed when another function is considered. In our implementation, we compute the RBF +quadrature weights by solving the linear system +(2.10) +� +Φ +P +P T +0 +� +� +�� +� +=A +� +w +v +� += +� +mRBF +mpoly +� +, +where v ∈ RK is a Lagrange multiplier3. Furthermore, the vectors mRBF ∈ RN and mpoly ∈ RK re- +spectively contain the moments of the translated kernels and polynomial basis functions: +mRBF = +� +I[ϕ1], . . . , I[ϕN] +�T , +mpoly = +� +I[p1], . . . , I[pK] +�T , +3The solution of (2.10) can be interpreted as the solution of an equality constrained linear optimization problem [5], +where v plays the role of a Lagrange multiplier. + +TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS +5 +with ϕn(x) = ϕ(εn∥x − xn∥2). The moments of different RBFs can be found in the appendix A. The +polynomial moments can be found in the literature, e. g., [37, Appendix A] and [29, 61]. +3. Stability and the Lebesgue constant. This section addresses the stability of RBF interpolants +and the corresponding RBF-QFs. +In particular, we show that both can be estimated in terms of +the Lebesgue constant. This was also observed in [32] for RBF-QFs on certain (compact) manifolds. +That said, we also demonstrate that RBF-QFs often come with improved stability compared to RBF +interpolation. +3.1. Stability of quadrature formulas. We shall start by addressing stability of RBF-QFs. To this +end, let us denote the best approximation of f from SN,d in the L∞-norm by ˆs. That is, +ˆs = arg min +s∈SN,d +∥f − s∥L∞(Ω) +with +∥f − s∥L∞(Ω) = sup +x∈Ω +|f(x) − s(x)|. +Note that this best approximation w. r. t. the L∞-norm is not necessarily equal to the RBF interpolant. +Still, the following error bound holds for the RBF-QF (2.9), that corresponds to exactly integrating the +RBF interpolant from SN,d: +(3.1) +|CN[f] − I[f]| ≤ +� +∥I∥∞ + ∥CN∥∞ +� +inf +s∈SN,d∥f − s∥L∞(Ω) +Inequality (3.1) is commonly known as the Lebesgue inequality; see, e. g., [94] or [9, Theorem 3.1.1]. It +is often encountered in polynomial interpolation [11, 49] but straightforwardly carries over to numerical +integration. In this context, the operator norms ∥I∥∞ and ∥CN∥∞ are respectively given by ∥I∥∞ = I[1] +and +∥CN∥∞ = +N +� +n=1 +|wn| = +N +� +n=1 +|I[cn]|. +Recall that the cn’s are the cardinal functions (see §2.1). In fact, ∥CN∥∞ is a common stability measure +for QFs. This is because the propagation of input errors, e. g., due to noise or rounding errors, can be +bounded by ∥CN∥∞: Let ˜f : Ω → R be a perturbed version of f, e. g. including noise or measurement +errors, then +|CN[f] − CN[ ˜f]| ≤ ∥CN∥∞∥f − ˜f∥L∞. +In other words, input errors are amplified at most by a factor that is equal to the operator norm ∥CN∥∞. +At the same time, we have +∥CN∥∞ ≥ CN[1], +where equality holds if and only if all quadrature weights are nonnegative. Also, for this reason, the +construction of QFs is mainly devoted to nonnegative QFs. +Definition 3.1 (Stability). +We call the RBF-QF CN stable if ∥CN∥∞ = CN[1]. This is the case if +and only if I[cn] ≥ 0 for all cardinal functions cn, n = 1, . . . , N. +It is also worth noting that CN[1] = ∥I∥∞ if the QF is exact for constants. For RBF-QFs, this is +the case if at least constants are included in the underlying RBF interpolant (d ≥ 0). + +6 +J. GLAUBITZ AND J. A. REEGER +3.2. Stability of RBF approximations. We now demonstrate how stability of the RBF-QF CN can +be connected to stability of the corresponding RBF interpolant. Indeed, the stability measure ∥CN∥∞ +can be bounded from above by +∥CN∥∞ ≤ ∥I∥∞ΛN, +with +ΛN := sup +x∈Ω +N +� +n=1 +|cn(x)|. +Here, ΛN is the Lebesgue constant corresponding to the recovery process f �→ sN,df (RBF interpolation). +Obviously, ΛN ≥ 1. Also note that if 1 ∈ SN,d (the RBF-QF is exact for constants), we observe +(3.2) +∥I∥∞ ≤ ∥CN∥∞ ≤ ∥I∥∞ΛN. +Hence, the RBF-QF is stable (∥CN∥∞ = ∥I∥∞) if ΛN is minimal (ΛN = 1). We briefly note that the +inequality ∥CN∥∞ ≤ ∥I∥∞ΛN is sharp by considering the following example. +Example 3.2 (∥CN∥∞ = ΛN). Let us consider the domain Ω = [0, 1] with ω ≡ 1, which immediately +implies ∥I∥∞ = 1. In [7] it was shown that for the linear PHS ϕ(r) = r and data points 0 = x1 < +· · · < xN = 1 the corresponding cardinal functions cm are simple hat functions. In particular, cm is +the ordinary “connect the dots” piecewise linear interpolant of the data pairs (xn, δnm), n = 1, . . . , N. +Thus, ΛN = 1. At the same time, this yields ∥CN∥∞ = 1 and therefore ∥CN∥∞ = ΛN. +Looking for minimal Lebesgue constants is a classical problem in approximation and recovery theory +[65, 92]. For instance, it is well known that for polynomial interpolation, even near-optimal sets of data +points yield a Lebesgue constant that grows as O(log N) in one dimension and as O(log2 N) in two +dimensions; see [11, 6, 8, 49]. In the case of RBF interpolation, the Lebesgue constant and appropriate +data point distributions were studied, for instance, in [50, 22, 64, 23]. That said, the second inequality +in (3.2) also tells us that in some cases, we can expect the RBF-QF to have superior stability properties +compared to the underlying RBF interpolant. Finally, it should be stressed that (3.2) only holds if +1 ∈ SN,d. In general, we have +CN[1] ≤ ∥CN∥∞ ≤ ∥I∥∞ΛN. +Still, this indicates that a recovery space SN,d is desired that yields a small Lebesgue constant as well +as the RBF-QF potentially having superior stability compared to RBF interpolation. +4. Compactly supported radial basis functions. Despite the increased use of RBF-QFs in appli- +cations, provable stability results are rarely encountered in the literature. As a first step towards a +more mature stability theory, we next prove stability of RBF-QFs for compactly supported kernels with +nonoverlapping supports. To be more precise, we subsequently consider RBFs ϕ : R+ +0 → R satisfying +the following restrictions: +(R1) ϕ is nonnegative, i. e., ϕ ≥ 0. +(R2) ϕ is uniformly bounded. W. l. o. g. we assume maxr∈R+ +0 |ϕ(r)| = 1. +(R3) ϕ is compactly supported. W. l. o. g. we assume supp ϕ = [0, 1]. +Already note that (R3) implies suppϕn = Bε−1 +n (xn), where +Bε−1 +n (xn) := { x ∈ Ω | ∥xn − x∥2 ≤ ε−1 +n }, +ϕn(x) := ϕ(εn∥xn − x∥2). +The ϕn’s will have nonoverlapping support if the shape parameters εn are sufficiently large. This can +be ensured by the following condition: +(4.1) +ε−1 +n +≤ hn := min +� +∥xn − xm∥2 | xm ∈ X \ {xn} +� +, +n = 1, . . . , N + +TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS +7 +Here, X denotes the set of data points. Finally, it should be pointed out that throughout this section, +we assume ω ≡ 1. This assumption is made for the main result, Theorem 4.1, to hold. Its role will +become clearer after consulting the proof of Theorem 4.1 and is revisited in Remark 4.8. +4.1. Main result. Our main result is the following Theorem 4.1. After collecting a few preliminary +results, its proof is given in §4.4. +Theorem 4.1. Let (xn)n∈N be an equidistributed sequence in Ω and XN = {xn}N +n=1. Furthermore, let +ω ≡ 1, let ϕ : R+ +0 → R be a RBF satisfying (R1) to (R3), and choose the shape parameters εn such that +the corresponding functions ϕn have nonoverlapping support and equal moments (I[ϕn] = I[ϕm] for all +n, m = 1, . . . , N). For every polynomial degree d ∈ N there exists an N0 ∈ N such that for all N ≥ N0 +the corresponding RBF-QF (2.9) is stable. That is, I[cm] ≥ 0 for all m = 1, . . . , N. +Note that a sequence (xn)n∈N is equidistributed in Ω if and only if +lim +N→∞ +|Ω| +N +N +� +n=1 +g(xn) = +� +Ω +g(x) dx +holds for all measurable bounded functions g : Ω → R that are continuous almost everywhere (in the +sense of Lebesgue), see [99]. For details on equidistributed sequences, we refer to the monograph [59].4 +Still, it should be noted that equidistributed sequences are dense sequences with a special ordering. In +particular, if (xn)n∈N ⊂ Ω is equidistributed, then for every d ∈ N there exists an N0 ∈ N such that +XN is Pd(Ω)-unisolvent for all N ≥ N0; see [38]. This ensures that the corresponding RBF interpolant +is well-defined. It should also be noted that if Ω ⊂ RD is bounded and has a boundary of measure zero +(again in the sense of Lebesgue), then an equidistributed sequence in Ω is induced by an equidistributed +sequence in the D-dimensional hypercube. More details on how an equidistributed sequence in Ω can +be constructed are provided in [38]. +Remark 4.2. It is always possible to ensure the equal moment condition, I[ϕn] = I[ϕm] for all +n, m = 1, . . . , N, in Theorem 4.1 by allowing the points closer to the boundary to come with a smaller +shape parameter. In this way, one can compensate for the part of the support cut off by the boundary +of the domain Ω. For instance, if equally spaced points are used on [a, b] with a = x1 < · · · < xN = b, +then the shape parameter is εn = ε for the interior points (n = 2, . . . , N − 1) and ε1 = εN = ε/2 for the +boundary points, where ε is a suitable chosen reference parameter. That said, in our numerical tests, +we observed Theorem 4.1 also to hold when the equal moment condition was not satisfied. +Remark 4.3. It is not necessary to include polynomials in the RBF-QF (2.9) for Theorem 4.1 to +imply stability. Indeed, it is subsequently proved by Lemma 4.5 that the RBF-QF (2.9) can also be +stable when no polynomials are included. +Sometimes, the RBF-QF (2.9) is also referred to as the +“RBF+poly-QF” when polynomials are included. In this regard, Theorem 4.1 shows that stability of +RBF-QFs carries over to RBF+poly-QFs under the assumptions listed in Theorem 4.1. The influence +of including polynomials into the RBF-QFs on their stability is also discussed for other kernels in §6. +4.2. Explicit representation of the cardinal functions. In preparation of proving Theorem 4.1 we +derive an explicit representation for the cardinal functions cn under the restrictions (R1)–(R3) and +(4.1). In particular, we use the concept of discrete orthogonal polynomials (DOPs). Let us define the +4Examples for equidistributed sequences include low-discrepancy points [47, 71, 14, 24] used in quasi-Monte Carlo +methods, such as the Halton points [44]. + +8 +J. GLAUBITZ AND J. A. REEGER +following discrete inner product corresponding to the data points XN = {xn}N +n=1: +(4.2) +[u, v]XN = |Ω| +N +N +� +n=1 +u(xn)v(xn) +Recall that the data points XN are coming from an equidistributed sequence and are ensured to be +Pd(Ω)-unisolvent for any degree d ∈ N if N is sufficiently large. In this case, (4.2) is positive definite +on Pd(Ω), i. e., [u, u]XN > 0 if u ∈ Pd(Ω) and u ̸= 0. We say that the basis {pk}K +k=1 of Pd(Ω), where +K = dim Pd(Ω), consists of DOPs if +[pk, pl]XN = δkl := +� +1 +if k = l, +0 +otherwise, +k, l = 1, . . . , K. +We now come to the desired explicit representation for the cardinal functions cm. +Lemma 4.4 (Explicit representation for cm). +Let the RBF ϕ : R+ +0 → R satisfy (R2) and (R3). Fur- +thermore, choose the shape parameters εn such that the corresponding functions ϕn have nonoverlapping +support and let the basis {pk}K +k=1 consists of DOPs. Then, the cardinal function cm, m = 1, . . . , N, is +given by +(4.3) +cm(x) = ϕm(x) − |Ω| +N +N +� +n=1 + + +K +� +k=1 +pk(xm)pk(xn) + + ϕn(x) + |Ω| +N +K +� +k=1 +pk(xm)pk(x). +Proof. Let m, n ∈ {1, . . . , N}. The restrictions (R2), (R3) together with the assumption of the ϕn’s +having nonoverlapping support yields ϕn(xm) = δmn. Hence, (2.6) and (2.7) imply +(4.4) +α(m) +n += δmn − +K +� +k=1 +β(m) +k +pk(xn). +If we substitute (4.4) into (2.3), we get +pl(xm) − N +|Ω| +K +� +k=1 +β(m) +k +[pk, pl]XN = 0, +l = 1, . . . , K. +Thus, if {pk}K +k=1 consists of DOPs, this gives us +(4.5) +β(m) +l += N +|Ω|pl(xm), +l = 1, . . . , K. +Finally, substituting (4.5) into (4.4) yields +α(m) +n += δmn − N +|Ω| +K +� +k=1 +pk(xm)pk(xn), +and therefore the assertion. +It should be stressed that using a basis of DOPs is not necessary for implementing RBF-QFs. In fact, +the quadrature weights are—ignoring computational considerations—independent of the polynomial +basis w. r. t. which the matrix P and the corresponding moments mpoly are formulated. We only use +DOPs as a theoretical tool to show stability of RBF-QFs. + +TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS +9 +4.3. Some low hanging fruits. Using the explicit representation (4.3) it is trivial to prove stability +of RBF-QFs (I[cm] ≥ 0 for all m = 1, . . . , N) when no polynomial term or only a constant is included +in the RBF interpolant. +Lemma 4.5 (No polynomials). +Let the RBF ϕ : R+ +0 → R satisfy (R1) to (R3) and choose the shape +parameters εn such that the corresponding functions ϕn have nonoverlapping support. Assume that no +polynomials are included in the corresponding RBF interpolant (K = 0). Then, the associated RBF-QF +is stable. +Proof. It is obvious that cm(x) = ϕm(x). Thus, by restriction (R1), cm is nonnegative and therefore +I[cm] ≥ 0. +Lemma 4.6 (Only a constant). Let the RBF ϕ : R+ +0 → R satisfy (R1) to (R3) and choose the shape +parameters εn such that the corresponding functions ϕn have nonoverlapping support. Assume that only +a constant is included in the corresponding RBF interpolant (K = 1). Then, the associated RBF-QF is +stable. +Proof. Let m ∈ {1, . . . , N}. If we choose p1 ≡ |Ω|−1/2, Lemma 4.4 yields +cm(x) = ϕm(x) + 1 +N + +1 − +N +� +n=1 +ϕn(x) + + . +Note that by (R2), (R3), and (4.1), we therefore have cm(x) ≥ ϕm(x). +Hence, (R1) implies the +assertion. +4.4. Proof of the main results. The following technical Lemma will be convenient to the proof of +Theorem 4.1. +Lemma 4.7. Let (xn)n∈N be equidistributed in Ω, XN = {xn}N +n=1, and let [·, ·]XN be the discrete inner +product (4.2). Furthermore, let {p(N) +k +}K +k=1 be a basis of Pd(Ω) consisting of DOPs w. r. t. [·, ·]XN . Then, +for all k = 1, . . . , K, +p(N) +k +→ pk +in L∞(Ω), +N → ∞, +where {pk}K +k=1 is a basis of Pd(Ω) consisting of continuous orthogonal polynomials satisfying +� +Ω +pk(x)pl(x) dx = δkl, +k, l = 1, . . . , K. +Moreover, it holds that +lim +N→∞ +� +Ω +p(N) +k +(x)p(N) +l +(x) dx = δkl, +k, l = 1, . . . , K. +Proof. The assertion is a combination of Lemma 11 and 12 from [37], where a general positive weight +function ω was considered. Here, we only consider the case ω ≡ 1. +Essentially, Lemma 4.7 states that if a sequence of discrete inner products converges to a continuous +one, then also the corresponding DOPs—assuming that the ordering of the elements does not change— +converges to a basis of continuous orthogonal polynomials. Furthermore, this convergence holds in a +uniform sense. We are now able to provide a proof for Theorem 4.1. + +10 +J. GLAUBITZ AND J. A. REEGER +Proof of Theorem 4.1. Let d ∈ N and m ∈ {1, . . . , N}. Under the assumptions of Theorem 4.1, we +have I[ϕn] = I[ϕm] for all n = 1, . . . , N. Thus, Lemma 4.4implies +I[cm] = I[ϕm] + +1 − |Ω| +N +N +� +n=1 +K +� +k=1 +p(N) +k +(xm)p(N) +k +(xn) + + + |Ω| +N +K +� +k=1 +p(N) +k +(xm)I[pk]. +Let {p(N) +k +}K +k=1 be a basis of Pd(Ω) consisting of DOPs. That is, [p(N) +k +, p(N) +l +]XN = δkl. In particular, +p(N) +1 +≡ |Ω|−1/2. With this in mind, it is easy to verify that +(4.6) +|Ω| +N +N +� +n=1 +K +� +k=1 +p(N) +k +(xm)p(N) +k +(xn) = +K +� +k=1 +p(N) +k +(xm)|Ω|1/2[p(N) +k +, p(N) +1 +]XN = 1. +Thus, we have +I[cm] ≥ 0 ⇐⇒ +K +� +k=1 +p(N) +k +(xm)I[p(N) +k +] ≥ 0. +Finally, observe that +K +� +k=1 +p(N) +k +(xm)I[p(N) +k +] = |Ω|1/2 +K +� +k=1 +p(N) +k +(xm) +� +Ω +p(N) +k +(x)p(N) +1 +(x) dx, +under the assumption that ω ≡ 1. Lemma 4.7 therefore implies +(4.7) +lim +N→∞ +K +� +k=1 +p(N) +k +(xm)I[p(N) +k +] = 1, +which completes the proof. +Remark 4.8 (On the assumption that ω ≡ 1). +The assumption that ω ≡ 1 in Theorem 4.1 is +necessary for (4.6) and (4.7) to both hold true. On the one hand, (4.6) is ensured by the the DOPs being +orthogonal w. r. t. the discrete inner product (4.2). This discrete inner product can be considered as an +approximation to the continuous inner product ⟨u, v⟩ = +� +Ω u(x)v(x) dx. This also results in Lemma4.7. +On the other hand, in general, (4.7) only holds if the DOPs converge to a basis of polynomials that +is orthogonal w. r. t. the weighted continuous inner product ⟨u, v⟩ω = +� +Ω u(x)v(x)ω(x) dx. Hence, for +(4.6) and (4.7) to both hold true at the same time, we have to assume that ω ≡ 1. In this case, the two +continuous inner products are the same. +5. Provable stable least squares RBF-QFs. Theorem 4.1 shows that compactly supported RBFs +(e. g. Wendland’s kernels) can lead to stable interpolatory QFs if the shape parameter is so that none +of the shifted kernels have a region of overlap. In our numerical tests, we observed this condition not +just to be sufficient but also often being “close to” necessary. We often found the RBF-QF even to have +negative weights when the support regions only slightly overlapped. At the same time, it is known that +scaling Wendland’s kernels so that the support decreases with the number of data points results in the +interpolation error to decrease only slowly or even to stagnate [27]. +To provide a more practical procedure for ensuring stability of RBF-QFs, we now demonstrate how +a least-squares approach [48, 42, 36, 37] can be used to construct stable RBF-QFs by allowing the + +TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS +11 +number of data points used for numerical integration to be larger than the number of centers that are +used to generate the RBF approximation space. The subsequent least-squares approach is not limited +to compactly supported kernels and can be used to construct stable QFs that are exact for fairly general +RBF approximation spaces. The only restrictions are that the RBF approximation space consists of +continuous and bounded functions and contains constants. Further, the number of data points used by +quadrature has to be sufficiently larger than the dimension of the RBF approximation space. Although +the least-squares approach has recently been extended to general multi-dimensional function spaces that +include constants in [38], the implications for RBF-QF have not yet been explored. To this end, we +consider a given center point set YM = {ym}M +m=1, generating the M-dimensional RBF space SM,d, and +a larger data point set XN = {xn}N +n=1 with N > M. Then, any QF CN[f] = �N +n=1 wnf(xn) that is +exact for all f ∈ SM,d has to satisfy +(5.1) + + +b1(x1) +. . . +b1(xN) +... +... +bM(x1) +. . . +bM(xN) + + +� +�� +� +=B + + +w1 +... +wN + + +� �� � +=w += + + +I[b1] +... +I[bM] + + +� +�� +� +=m +, +where {bm}M +m=1 is a basis of SM,d. The matrix B in (5.1) depends on XN and YM (as well as on the +kernel ϕ and the polynomial degree d), which we denote by B = B(XN, YM). Assume that the data +point set XN is SM,d-unisolvent, i. e., +f(xn) = 0, ∀xn ∈ XN =⇒ f ≡ 0 +holds for all f ∈ SM,d.5 Then (5.1) has infinitely many solutions, which form a (N − M)-dimensional +affine linear subspace W. Every w ∈ W yields a QF that is exact for all functions from the RBF space +SM,d. We want to find a positive solution w ∈ W so that the corresponding QF with weights w is stable +(see Definition 3.1). To this end, we use the following result from [38]. +Lemma 5.1 (Corollary 3.6 in [38]). Let Ω ⊂ RD, ω : Ω → R+ +0 be a Riemann integrable weight function +that is positive almost everywhere, and let F = span{ bm | m = 1, . . . , M } be a finite-dimensional +linear space of continuous and bounded functions that contains constants. Further, let (xn)n∈N be an +equidistributed sequence in Ω with ω(xn) > 0 for all n ∈ N and denote the affine linear subspace of +solutions of (5.1) by WF. Then there exists an N0 ∈ N such that for all N ≥ N0 and discrete weights +rn,N = |Ω|ω(xn)/N, +n = 1, . . . , N, +the corresponding least-squares QF +CLS +N [f] = +N +� +n=1 +wLS +n f(xn) +with +wLS = arg min +w∈WF +∥R−1/2w∥2, +where R−1/2 = diag(1/√r1, . . . , 1/√rN), is positive and exact for all f ∈ F. +If we apply Lemma 5.1 to the RBF function space SM,d, we get Corollary 5.2. +5XN is SM,d-unisolvent, for instance, when the kernel ϕ is conditionally positive definite of order d and XN is Pd(Ω)- +unisolvent, which is a common assumption to ensure uniqueness of RBF interpolants. + +12 +J. GLAUBITZ AND J. A. REEGER +Corollary 5.2. Let Ω ⊂ RD be compact and let ω : Ω → R+ +0 be a Riemann integrable weight function +that is positive almost everywhere. Further, let d ≥ 0 be an integer, let ϕ : R+ +0 :→ R be a continuous +and conditionally positive kernel of order d, and let {ym}M +m=1 be a given set of centers. If (xn)n∈N is +an equidistributed sequence in Ω with ω(xn) > 0 for all n ∈ N, then there exists an N0 ∈ N such that +for all N ≥ N0 and discrete weights +rn,N = |Ω|ω(xn)/N, +n = 1, . . . , N, +the corresponding least-squares RBF-QF +(5.2) +CLS +N [f] = +N +� +n=1 +wLS +n f(xn) +with +wLS = arg min +w∈W +∥R−1/2w∥2, +where R−1/2 = diag(1/√r1, . . . , 1/√rN), is positive and exact for all f ∈ SM,d. +Proof. We first note that the RBF function space SM,d, which we defined in §2, is M-dimensional +with M < ∞, i. e., finite-dimensional. Because the kernel ϕ and all polynomials up to degree d are +continuous, all functions from SM,d are continuous. Further, since Ω ⊂ RD is compact and all functions +from SM,d are continuous, they are also bounded. Finally, d ≥ 0 implies that SM,d contains constants. +Corollary 5.2 now follows from Lemma 5.1 with F = SM,d. +The weighted least-squares solution (5.2) has the advantage of being easy and efficient to compute +using standard tools from linear algebra. The above discussion motivates us to formulate the following +procedure to construct stable least-squares RBF-QFs (LSRBF-QFs). +Algorithm 5.1 Constructing stable LSRBF-QFs +1: Input: Center points {ym}M +m=1, kernel ϕ, polynomial degree d ≥ 0, weight function ω, and equidis- +tributed data points (xn)n∈N +2: Output: An integer N ≥ M and a stable LSRBF-QF with points {xn}N +n=1 and weights wLS ∈ RN +3: Set wLS equal to the weights of the interpolatory RBF-QF given by (2.10) +4: repeat +5: +Increase the number of data points by one: N = N + 1 +6: +Set XN = {xn}N +n=1 +7: +Compute the matrix B = B(XN, YM) as in (5.1) +8: +Compute the weighted least-squares solution wLS as in (5.2) +9: +Determine the smallest weight: wmin = min(wLS) +10: until wLS ≥ 0 +Algorithm 5.1 assumes that XM is SM,d-unisolvent, since the interpolatory RBF-QF given by (2.10) +would not be defined otherwise. The possible advantage of stable LSRBF-QFs compared to (potentially +unstable) interpolatory RBF-QFs is demonstrated in §7.2. Finally, we point out the potential application +of stable LSRBF-QFs to the construction of stable RBF methods for time-dependent hyperbolic partial +differential equations [90, 41]. A crucial part of these methods is replacing exact integrals involving +the approximate solution—in this case, an (local) RBF function— with a quadrature that should be as +accurate as possible for functions from the approximation space. +6. Polynomial terms do not influence asymptotic stability. Recall that Theorem 4.1 in §4 holds +regardless of the degree d of the polynomial term included in the RBF interpolant. Indeed, one might +generally ask, “how are polynomial terms influencing stability of the RBF-QF?”. In what follows, we + +TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS +13 +address this question by showing that—under certain assumptions that are to be specified yet—at least +asymptotic stability of RBF-QFs is independent of polynomial terms. +Recently, the following explicit formula for the cardinal functions was derived in [5, 4]. +Let us +denote c(x) = [c1(x), . . . , cN(x)]T , where c1, . . . , cN are the cardinal functions spanning SN,d; see (2.6) +and (2.7). Provided that Φ and P in (2.5) have full rank6, +(6.1) +c(x) = ˆc(x) − Bτ(x) +holds. +Here, ˆc(x) = [ˆc1(x), . . . , ˆcN(x)]T are the cardinal functions corresponding to the pure RBF +interpolation without polynomials. That is, they span SN,−1. At the same time, B and τ are defined +as +B := Φ−1P +� +P T Φ−1P +�−1 +, +τ(x) := P T ˆc(x) − p(x) +with p(x) = [p1(x), . . . , pK(x)]T . Note that τ can be interpreted as a residual measuring how well pure +RBFs can approximate polynomials up to degree d. Recalling (2.9), we see that (6.1) implies +(6.2) +w = ˆw − BI[τ], +where w is the vector of quadrature weights of the RBF-QF with polynomials (d ≥ 0). At the same +time, ˆw is the vector of weights corresponding to the pure RBF-QF without polynomial augmentation +(d = −1). Moreover, I[τ] denotes the componentwise application of the integral operator I. It was +numerically demonstrated in [5] that for fixed d ∈ N one has +(6.3) +max +x∈Ω ∥Bτ(x)∥ℓ∞ → 0 +as +N → ∞ +if PHS are used. Note that, for fixed x ∈ Ω, Bτ(x) is an N-dimensional vector and ∥Bτ(x)∥ℓ∞ denotes +its ℓ∞-norm. That is, the maximum absolute value of the N components. It should be pointed out that +(6.3) was numerically demonstrated only for PHS in [5]. However, the relations (6.1) and (6.2) hold for +general RBFs as well as varying shape parameters, assuming that Φ and P have full rank. Please see +[5, Section 4] for more details. We also remark that (6.3) implies the weaker statement +(6.4) +∥Bτ(·)∥ℓ1 → 0 in L1(Ω) +as +N → ∞. +Here, Bτ(·) denotes a vector-valued function, Bτ : Ω → RN. That is, for a fixed argument x ∈ Ω, +Bτ(x) is an N-dimensional vector in RN and ∥Bτ(x)∥ℓ1 denotes the usual ℓ1-norm of this vector. Thus, +(6.4) means that the integral of the ℓ1-norm of the vector-valued function Bτ(·) converges to zero as +N → ∞. The above condition is not just weaker than (6.3) (see Remark 6.4), but also more convenient +to investigate stability of QFs. Indeed, we have the following results. +Theorem 6.1. Let ω ∈ L∞(Ω). Assume Φ and P in (2.5) have full rank, and assume (6.4) holds. +Then the two following statements are equivalent: +(a) ∥ ˆw∥ℓ1 → ∥I∥∞ for N → ∞ +(b) ∥w∥ℓ1 → ∥I∥∞ for N → ∞ +That is, either both the pure and polynomial augmented RBF-QF are asymptotically stable or none is. +A short discussion on the term “asymptotically stable” is subsequently provided in Remark 6.2. +6P having full rank means that P has full column rank, i. e., the columns of P are linearly independent. This is +equivalent to the set of data points being Pd(Ω)-unisolvent. + +14 +J. GLAUBITZ AND J. A. REEGER +Proof. Assume Φ and P in (2.5) have full rank, and assume (6.4) holds. Then (6.2) follows and +therefore +(6.5) +∥w∥ℓ1 ≤ ∥ ˆw∥ℓ1 + ∥BI[τ]∥ℓ1, +∥ ˆw∥ℓ1 ≤ ∥w∥ℓ1 + ∥BI[τ]∥ℓ1. +Next, note that BI[τ] = I[Bτ], and thus +∥BI[τ]∥ℓ1 = +N +� +n=1 +��I[(Bτ)n] +�� ≤ I + + +N +� +n=1 +|(Bτ)n| + + = I +� +∥Bτ∥ℓ1 +� +. +Since ω ∈ L∞(Ω), it follows that +∥BI[τ]∥ℓ1 ≤ ∥ω∥L∞(Ω) +� +Ω +∥Bτ(x)∥ℓ1 dx. +Hence, by assuming that (6.4) holds, we get ∥BI[τ]∥ℓ1 → 0 for fixed d ∈ N and N → ∞. Finally, +substituting this into (6.5) yields the assertion. +Theorem 6.1 states that–under the listed assumptions—it is sufficient to consider asymptotic stabil- +ity of the pure RBF-QF. Once asymptotic (in)stability is established for the pure RBF-QF, by Theorem +6.1, it also carries over to all corresponding augmented RBF-QFs. Interestingly, this follows our find- +ings for compactly supported RBFs reported in Theorem 4.1. There, conditional stability was ensured +independently of the degree of the augmented polynomials. +Remark 6.2 (Asymptotic stability). +We call a sequence of QFs with weights wN ∈ RN for N ∈ N +asymptotically stable if ∥wN∥ℓ1 → ∥I∥∞ for N → ∞. Recall that ∥wN∥ℓ1 = ∥CN∥∞ if the weights wN +correspond to the N-point QF CN. It is easy to note that this is a weaker property than every single +QF being stable, i. e., ∥wN∥ℓ1 = ∥I∥∞ for all N ∈ N. That said, consulting (3.1), asymptotic stability +is sufficient for the QF to converge for all functions that can be approximated arbitrarily accurate by +RBFs w. r. t. the L∞(Ω)-norm. Of course, the propagation of input errors might be suboptimal for every +single QF. +Theorem 6.1 makes two assumptions. (1) Φ and P are full rank matrices; and (2) the condition +(6.3) holds. In the two following remarks, we comment on these assumptions. +Remark 6.3 (On the first assumption of Theorem 6.1). +Although requiring A and P to have full +rank might seem restrictive, there are often even more restrictive constraints in practical problems. For +instance, when solving partial differential equations, the data points are usually required to be smoothly +scattered so that the distance between data points is kept roughly constant. It seems unlikely to find +A and P to be singular for such data points. See [5] for more details. +Remark 6.4 (On the second assumption of Theorem 6.1). +The second assumption for Theorem 6.1 +to hold is that (6.4) is satisfied. That is, the integral of ∥Bτ(·)∥ℓ1 : Ω → R+ +0 converges to zero as +N → ∞. This is a weaker condition than the maximum value of ∥Bτ(·)∥ℓ1 converging to zero, which +was numerically observed to hold for PHS in [5]. The relation between these conditions can be observed +by applying H¨older’s inequality (see, for instance, [81, Chapter 3]). Let 1 ≤ p, q ≤ ∞ with 1/p+1/q = 1 +and assume that ω ∈ Lq(Ω). Then we have +� +Ω +∥Bτ(x)∥ℓ1ω(x) dx ≤ +�� +Ω +∥Bτ(x)∥p +ℓ1 dx +�1/p �� +Ω +ω(x)q dx +�1/q +. + +TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS +15 +100 +102 +10-2 +100 +102 +1/h +(a) k = 1, d = −1 (pure RBF) +100 +102 +100 +105 +1/h +(b) d = 0 (constant term) +100 +102 +100 +105 +1/h +(c) d = 1 (linear term) +Figure 1: The stability measure ∥CN∥∞ for Wendland’s compactly supported RBF ϕ1,k with smoothness param- +eters k = 0, 1, 2 on N = 100 equidistant data points. 1/h denotes the threshold above which the basis functions +have nonoverlapping support. +Hence, ∥Bτ∥ℓ1 converging to zero in Lp(Ω) as N → ∞ for some p ≥ 1 immediately implies (6.2). The +special case of p = ∞ corresponds to (6.3). +7. Numerical results. We present a variety of numerical tests in one and two dimensions to demon- +strate our theoretical findings. A constant weight function ω ≡ 1 is used for simplicity. All numerical +tests presented here were generated in MATLAB7. +7.1. Compactly supported RBFs. Let us start with demonstrating Theorem 4.1 in one dimension. +To this end, we consider Wendland’s compactly supported RBFs in Ω = [0, 1]. +Figure 1 illustrates the stability measure ∥CN∥∞ of Wendland’s compactly supported RBF ϕ1,k +with smoothness parameters k = 0, 1, 2 as well as the optimal stability measure. The latter is given by +CN[1] if no constants are included and by ∥I∥∞ = 1 if constants are included in the RBF approximation +space. Furthermore, N = 100 equidistant data points were used, including the end points, x1 = 0 +and xN = 1, and the (reference) shape parameter ε was allowed to vary. Finally, 1/h denotes the +threshold above which the compactly supported RBFs have nonoverlapping support. We note that the +RBF-QFs are stable for sufficiently small shape parameters. At the same time, we can also observe +the RBF-QF be stable for ε ≥ 1/h. It can be argued that this is in accordance with Theorem 4.1. +Recall that Theorem 4.1 essentially states that for ε ≥ 1/h, and assuming that all basis functions have +equal moments (I[ϕn] = I[ϕm] for all n, m), the corresponding RBF-QF (including polynomials of any +degree) is stable if a sufficiently large number of equidistribiuted data points is used. Here, the equal +moments condition was ensured by choosing the shape parameter as εn = ε for the interior data points +(n = 2, . . . , N − 1) and as ε1 = εN = ε/2 for the boundary data points. +That said, at least numerically, we observe that it is possible to drop this equal moment condition. +This is demonstrated by Figure 2. There, we perform the same test as in Figure 1, except choosing +all the shape parameters to be equal (εn = ε, n = 1, . . . , N) and going over to nonequidistant Halton +points. Nevertheless, we can see in Figure 2 that for ε ≥ 1/h the RBF-QFs are still stable. +Next, we extend our numerical tests to the following Genz test functions [33] (also see [94]) on +7See https://github.com/jglaubitz/stability RBF CFs + +16 +J. GLAUBITZ AND J. A. REEGER +100 +102 +100 +101 +1/h +(a) d = 0 (constant term) +100 +102 +100 +101 +102 +1/h +(b) d = 1 (linear term) +Figure 2: The stability measure ∥CN∥∞ for Wendland’s compactly supported RBF ϕ1,k with smoothness pa- +rameters k = 0, 1, 2 on N = 100 Halton points. 1/h denotes the threshold above which the basis functions have +nonoverlapping support. +Ω = [0, 1]q: +(7.1) +g1(x) = cos + +2πb1 + +q +� +i=1 +aixi + + , +g2(x) = +q +� +i=1 +� +a−2 +i ++ (xi − bi)2�−1 +, +g3(x) = + +1 + +q +� +i=1 +aixi + + +−(q+1) +, +g4(x) = exp + +− +q +� +i=1 +a2 +i (xi − bi)2 + + +Here, q denotes the dimension under consideration and is henceforth chosen as q = 2. These functions +are designed to have different complex characteristics for numerical integration routines. The vectors +a = (a1, . . . , aq)T and b = (b1, . . . , bq)T respectively contain (randomly chosen) shape and translation +parameters. For each case, the experiment was repeated 100 times. At the same time, for each experi- +ment, the vectors a and b were drawn randomly from [0, 1]2. For reasons of space, we only report the +results for g1 and k = 1 in Figure 3. As before, the smallest errors are found for shape parameters cor- +responding to the stable RBF-QF. The results for g2, g3, g4 and k = 0, 2 were similar and are therefore +not reported here. Since it might be hard to identify the smallest errors as well as the corresponding +shape parameter and stability measure from Figure 3, these are listed in Table 2 for d = 0, 1 together +with the corresponding values for the fourth Genz test function g4. +We see in Figure 3 that for d = −1 and increasing ε, the error increases. This is because the supports +of the translated kernels (disks in 2d) become smaller, resulting in “holes” in the pure RBF interpolant, +i. e., regions where it is zero. In Figure 3c, for random points and d = −1, the supports become so +small that all the quadrature weights become zero. The holes vanish if at least a constant is included +in the RBF interpolant, which explains the reduced errors for the same value of ε when d = 0 or d = 1. +Finally, even for nonoverlapping supports (ε = 1/h), the area of the holes in the pure RBF part of the +interpolant can converge to zero8 as N → ∞. +8Assuming the sequence of points is dense in Ω + +TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS +17 +10-1 +101 +103 +10-6 +10-2 +102 +1/h +(a) Equidistant, d = −1 +10-1 +101 +103 +10-6 +10-2 +102 +1/h +(b) Halton, d = −1 +10-1 +101 +103 +10-6 +10-2 +102 +1/h +(c) Random, d = −1 +10-1 +101 +103 +10-6 +10-2 +102 +1/h +(d) Equidistant, d = 0 +10-1 +101 +103 +10-6 +10-2 +102 +1/h +(e) Halton, d = 0 +10-1 +101 +103 +10-6 +10-2 +102 +1/h +(f) Random, d = 0 +10-1 +101 +103 +10-6 +10-2 +102 +1/h +(g) Equidistant, d = 1 +10-1 +101 +103 +10-6 +10-2 +102 +1/h +(h) Halton, d = 1 +10-1 +101 +103 +10-6 +10-2 +102 +1/h +(i) Random, d = 1 +Figure 3: Error analysis for Wendland’s compactly supported RBF ϕ2,k and the first Genz test function g1 on +Ω = [0, 1]2; see (7.1). In all cases, N = 400 data points (equidistant, Halton, or random) were considered. 1/h +denotes the threshold above which the basis functions have nonoverlapping support. +Remark 7.1. If the RBF interpolant sN,df convergences to f in L1(Ω) as N → ∞, we get +��CN[f] − I[f] +�� = +��I[sN,df] − I[f] +�� ≤ +� +Ω +|(sN,df)(x) − f(x)| dx → 0, +N → ∞, +and therefore CN[f] → I[f] as N → ∞. For convergence results of RBF interpolants, we refer to the +monographs [12, 98, 27]. That said, we point out that the convergence of sN,df to f depends on the area +not covered by the supports. Let us denote the area that is covered by the supports by Ωsupp(ϕ,XN,ε), +then the area that is not covered by the supports is Ω\Ωsupp(ϕ,XN,ε). A rough but simple lower bound for +the L1(Ω)-error of sN,df and f is as follows. Note that the RBF interpolant is zero on Ω \ Ωsupp(ϕ,XN,ε) + +18 +J. GLAUBITZ AND J. A. REEGER +g1 +g4 +emin +ε +∥CN∥∞ +emin +ε +∥CN∥∞ +Equidistant Points +d = 0 +2.2e-05 +2.6e+00 +1.0e+00 +6.1e-05 +2.6e+00 +1.0e+00 +d = 1 +2.2e-05 +2.6e+00 +1.0e+00 +6.2e-05 +2.6e+00 +1.0e+00 +Halton Points +d = 0 +4.7e-05 +5.5e-01 +1.0e+00 +1.9e-05 +5.5e-01 +1.0e+00 +d = 1 +1.1e-05 +5.5e-01 +1.0e+00 +1.6e-05 +5.5e-01 +1.0e+00 +Random Points +d = 0 +6.0e-04 +2.5e-01 +1.0e+00 +1.6e-04 +2.9e-01 +1.0e+00 +d = 1 +2.2e-04 +4.0e-01 +1.0e+00 +1.7e-04 +4.0e-01 +1.0e+00 +Table 2: Minimal errors, emin, for the first and fourth Genz test function, g1 and g4, together with the correspond- +ing shape parameter, ε, and stability measure, ∥CN∥∞. Wendland’s compactly supported RBF with smoothness +parameter k = 1 was used in all cases. +and thus +(7.2) +� +Ω +|(sN,df)(x) − f(x)| dx ≥ +� +Ω\Ωsupp(ϕ,XN,ε) +|f(x)| dx, +where the right-hand side is the average absolute value of f in the area not covered by the supports. +(7.2) indicates that convergence includes the rate with which “holes” go to zero. +In Figures 4 and 5, we relate the error in computing the integral of g1 on Ω = [0, 1]2 to the portion +of the domain that is not covered by the possibly overlapping supports of the Wendland functions. For +equidistant points, the horizontal/vertical distance between adjacent points is (i.e. xij and x(i±1)j or +xij and xi(j±1)) is 1/( +√ +N − 1), so as long as ε > 2( +√ +N − 1) the circles do not overlap and the total +area that is not covered by the supports is given by +1 − π +ε2 ( +√ +N − 1)2. +Once ε ≤ 2( +√ +N −1), the circles overlap, and the total area that is not covered by the supports becomes +1 − π +ε2 ( +√ +N − 1)2 + 2(θ − sin(θ)) +ε2 +( +√ +N − 1)2, +where +θ = 2sin−1 + + + +� +4( +√ +N − 1)2 − ε2 +2( +√ +N − 1) + + + . + +TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS +19 +-6 +-5 +-4 +-3 +-3 +-2 +-2 +-1 +-1 +-1 +0 +1 +2 +1.5 +2 +2.5 +3 +3.5 +-8 +-7 +-6 +-5 +-4 +-3 +-2 +-1 +0.2 +0.2 +0.4 +0.4 +0.6 +0.6 +0.8 +-1 +0 +1 +2 +1.5 +2 +2.5 +3 +3.5 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Figure 4: Error of CN[g1] on Ω = [0, 1]2 and the area not covered by the supports of the Wendland functions for +various N and ε when no constant is included in the RBF interpolant +-5 +-4 +-4 +-3 +-3 +-2 +-2 +-1 +0 +1 +2 +1.5 +2 +2.5 +3 +3.5 +-8 +-7 +-6 +-5 +-4 +-3 +-2 +-1 +0.2 +0.2 +0.4 +0.4 +0.6 +0.6 +0.8 +-1 +0 +1 +2 +1.5 +2 +2.5 +3 +3.5 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Figure 5: Error of CN[g1] on Ω = [0, 1]2 and the area not covered by the supports of the Wendland functions for +various N and ε when a constant is included in the RBF interpolant +Finally, when ε ≤ +√ +2( +√ +N −1), the area that is not covered by the supports is 0. Figure 4 illustrates the +error (left frame) and the area not covered by the supports (right frame) in this situation for various N +and ε. The dashed lines represent the cases ε = 2( +√ +N − 1) and ε = +√ +2( +√ +N − 1). On the other hand, +Figure 5 illustrates the same test for an RBF interpolant that includes a constant. It demonstrates the +improvement when the constant basis element covers the holes. + +20 +J. GLAUBITZ AND J. A. REEGER +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(a) Random points +101 +102 +103 +10-4 +10-3 +10-2 +10-1 +100 +(b) Minimal and maximal distance +Figure 6: Random points and the corresponding minimal and maximal distance, (7.3) and (7.4). The minimal +distance is the smallest distance between any two distinct points. The maximal distance is the largest distance +between any point and the closest distinct point. +7.2. Stable LSRBF-QFs. We demonstrate that the least-squares approach discussed in §5 can +stabilize RBF-QFs. We repeat that stable LSRBF-QF can be constructed for any RBF function space +as long as we are willing to oversample, i. e., the number of data points used by the quadrature is larger +than the dimension of the RBF function space. In other words, there are more data points than center +points. Notably, oversampling was used in some recent works [89, 90, 41] to stabilize RBF methods +for partial differential equations, and it would be of interest to combine this with LSRBF-QF in future +works. Here, we demonstrate the possible advantage of LSRBF-QFs compared to interpolatory RBF- +QFs (data and center points are the same) for the RBF function space spanned by a constant and +the functions ϕ(ε∥x − ym∥) on Ω = [0, 1]2 using a Gaussian kernel ϕ(r) = exp(−r2) and a constant +(independent of the center and data points) shape parameter ε = 0.8. The center and data points, +{ym}M +m=1 and {xn}N +n=1, are chosen as the first M and N elements of the same sequence of random +points, respectively. +Figure 6a illustrates the first 64 random points from the sequence used in our tests. Figure 6b +visualizes the minimal and maximal distance of the random data point set XN = {xn}N +n=1 for different +values for N. We define the minimal distance of XN, denoted by hmin(XN), as the smallest distance +between any two distinct points, +(7.3) +hmin(XN) = +min +xn∈XN +min +xm∈XN\xn ∥xm − xn∥2. +At the same time, we define the maximal (filling) distance of XN, denoted by hmax(XN), as the largest +distance between any point and the closest distinct point, +(7.4) +hmax(XN) = max +xn∈XN +min +xm∈XN\xn +∥xm − xn∥2. +Figure 7a provides the values of the stability measure for the interpolatory RBF-QF (“RBF”) and +the stable LSRBF-QF (“LSRBF”). For the same center points (same RBF function space for which +the quadrature is exact), the LSRBF-QF uses more data points to evaluate the integrand than the + +TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS +21 +101 +102 +103 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +1.5 +(a) Stability measure +101 +102 +103 +10-8 +10-6 +10-4 +10-2 +100 +(b) Errors, no noise +101 +102 +103 +10-6 +10-4 +10-2 +100 +(c) Errors, noise of magnitude 10−4 +101 +102 +103 +10-4 +10-3 +10-2 +10-1 +(d) Errors, noise of magnitude 10−2 +Figure 7: Stability measure and errors for Genz’ first test function g1 on Ω = [0, 1]2 with ω ≡ 1 using an +interpolatory RBF-QF and a stable LSRBF-QF. Random points and a Gaussian kernel with a constant shape +parameter were used. +interpolatory RBF-QF. The other way around, for the same data points, the interpolatory RBF-QF is +exact for a larger RBF function space than the LSRBF-QF. At the same time, the interpolatory RBF- +QF is found to have a suboptimal stability measure (due to negative weights), which results in stability +issues. In contrast, in all cases, the LSRBF-QF has an optimal stability measure (due to the weights +being positive). Further, Figure 7b reports on the errors of the interpolatory RBF-QF and the stable +LSRBF-QF on N random points applied to Genz’ first test function g1 on Ω = [0, 1]2 with ω ≡ 1. In +this example, both formulas perform similarly. In Figures 7c and 7d, we repeated this experiment but +added uniformly distributed noise of magnitude 10−4 and 10−2 to the function values at the data points. +The accuracy of the interpolatory RBF-QF deteriorates notably stronger than that of the LSRBF-QF +in the presence of noise due to the improved stability of the latter. We made the same observation also +for other point distributions and Genz test functions. +The LSRBF-QF having an optimal stability measure (being positive) for sufficiently large N can +be explained by Corollary 5.2 since we are given a compact domain, a positive weight function, and + +22 +J. GLAUBITZ AND J. A. REEGER +100 +101 +102 +101 +102 +103 +(a) Random points +100 +101 +102 +101 +102 +103 +(b) Halton points +Figure 8: The smallest number of random/Halton data points, N, needed to find a positive LSRBF-QF that is +exact for the RBF approximation space induced by the first M random/Halton center points. +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(a) Halton points +101 +102 +103 +10-2 +10-1 +100 +(b) Minimal and maximal distance +Figure 9: Halton points and the corresponding minimal and maximal distance, (7.3) and (7.4). The minimal +distance is the smallest distance between any two distinct points. The maximal distance is the largest distance +between any point and the closest distinct point. +a function space of continuous and bounded functions that contains constants. Figure 8 reports the +smallest number of random/Halton data points N we needed to find a positive LSRBF-QF that is exact +for the RBF approximation space induced by using the first M random/Halton points as centers. To +also illustrate the semi-random Halton points, Figure 9 visualizes the first 64 Halton points and their +minimal and maximal distance for an increasing number N. Considering the model “N = C · Ms” and +performing a least-squares fit for the parameters C and s given the data illustrated in Figure 8 revealed +the following: For random data and center points, we found C ≈ 2.1·10−1 and s ≈ 1.9. For Halton data +and center points, we found C ≈ 4.9·10−2 and s ≈ 2.1. In both cases, we found N to be roughly linearly + +TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS +23 +proportional to the squared dimension of the approximation space for which the positive least-squares +quadrature is exact. Similar ratios were also observed in [48, 42, 36, 35, 37, 16, 17, 66, 38]. It might +be argued that the observed ratio between N and M is necessary for the LSRBF-QF to avoid inherent +stability issues predicted by the ’impossibility’ theorem proved in [74], which states that any procedure +for approximating univariate functions from equally spaced samples that converges exponentially fast +must also be exponentially ill-conditioned. +Finally, we address the convergence rate observed in Figure 7b. In theory, Gaussian RBF inter- +polants can converge almost exponentially fast9 in the L∞(Ω) with the maximal (filling) distance. For +simplicity, we considered the model “|I[f] − CN[f]| = exp(−Chmax(XN)s)” and performing a least- +squares fit for the parameters C and s using the data presented in Figure 7b. For the interpolatory +RBF-QF, we found C ≈ 2.0 and s ≈ −1.1. For the LSRBF-QF, we found C ≈ 1.8 and s ≈ −1.3. Both +QFs converge roughly exponentially, with the LSRBF-QF converging slightly faster than the interpola- +tory RBF-QF, even in the noiseless case. A more general comment on the convergence of LSRBF-QFs +is offered in Remark 7.2. +Remark 7.2 (Convergence of LSRBF-QF). Assume that the positive LSRBF-QF CN on Ω is exact +for all functions from the RBF approximation space SM,d(Ω) with d ≥ 0. Due to CN being positive and +exact for constants, we have ∥CN∥∞ = ∥I∥∞ and the Lebesgue inequality (3.1) implies +|CN[f] − I[f]| ≤ 2∥I∥∞ +� +inf +s∈SM,d(Ω)∥f − s∥L∞(Ω) +� +for any continuous f : Ω → R. Now consider a sequence of positive LSRBF-QFs (CN)N∈N with CN +being exact for SM,d(Ω) with M = M(N). Assume that SM,d(Ω) ⊂ SM+1,d(Ω) for all M ∈ N and +that the given function f lies in � +M∈N SM,d(Ω). Thus, if M(N) → ∞ for N → ∞, then (CN[f])N∈N +converges to I[f] as N → ∞. Let us now assume that the ratio between M and N is of the form +N = O(M2), which we numerically observed to be true. The convergence rate of the LSRBF-QF for +f is then the square root of the convergence rate of the best approximation of f from the sequence of +approximation spaces (SM,d(Ω))N∈N for the L∞(Ω)-norm. +7.3. Polyharmonic splines. We end this section by providing a similar investigation for PHS. Again, +the first and fourth Genz test functions on Ω = [0, 1]2 are considered. However, no shape parameter +is involved for PHS, and we consider their stability and accuracy for an increasing number of Halton +points. Figure 10 shows the results for the cubic (ϕ(r) = r3) and quintic (ϕ(r) = r5) PHS RBF. We +either added no polynomials (d = −1) or polynomial terms of order d = 0 and d = 1. +Figure 10 +shows that all RBF-QFs converge while being stable or at least asymptotically stable, independent of +the added polynomial term. In particular, we see that adding polynomial terms does not affect the +asymptotic stability of the PHS RBF-QFs, per our results from §6. Finally, we see that the convergence +rate of the RBF-QF depends on the kernel rather than being governed by the polynomial degree d. +The added polynomial term reduces the error of the PHS RBF-QF but not the convergence rate. We +observe second-order convergence for the cubic PHS RBF and third-order convergence for the quintic +PHS RBF.10 +9For a function from the appropriate native function space, the L∞(Ω)-error between the function and its RBF inter- +polant is in O(exp(−C log hmax(XN)/hmax(XN))); see [98]. +10In Figure 10a the cubic PHS RBF-QF first shows third-order convergence before it then settles for second-order +convergence. We believe that the observed initial third-order decrease in the error is a combination of the second-order +approximation rate of the cubic PHS-RBF interpolant and the decreasing Lebesgue constant ∥CN∥∞ in (3.1). Once the +QF is stable (∥CN∥∞ = ∥I∥∞), the second-order approximation rate dominates the error of the QF, and we thus start to +observe second-order convergence. + +24 +J. GLAUBITZ AND J. A. REEGER +101 +102 +103 +10-10 +100 +(a) Cubic, d = −1 +101 +102 +103 +10-10 +100 +(b) Cubic, d = 0 +101 +102 +103 +10-10 +100 +(c) Cubic, d = 1 +101 +102 +103 +10-10 +100 +(d) Quintic, d = −1 +101 +102 +103 +10-10 +10-5 +100 +(e) Quintic, d = 0 +101 +102 +103 +10-10 +10-5 +100 +(f) Quintic, d = 1 +Figure 10: Error analysis for the cubic (ϕ(r) = r3) and quintic (ϕ(r) = r5) PHS RBF in two dimensions using +Halton points. The first and fourth Genz test functions g1, g4 were considered on Ω = [0, 1]2; see (7.1). +8. Concluding thoughts. In this work, we investigated stability of RBF-QFs. We started by show- +ing that stability of RBF-QFs can be connected to the famous Lebesgue constant of the underlying +RBF interpolant. This indicates that RBF-QFs might benefit from low Lebesgue constants. Further- +more, stability was proven for RBF-QFs based on compactly supported RBFs under the assumption +of a sufficiently large number of (equidistributed) data points and the shape parameter(s) lying above +a certain threshold. Finally, we showed that under certain conditions, asymptotic stability of RBF- +QFs is independent of polynomial terms included in RBF approximations. A series of numerical tests +accompanied the above findings. +Acknowledgements. JG was supported by AFOSR #F9550-18-1-0316 and ONR #N00014-20-1- +2595. +We thank Toni Karvonen for pointing out the connection between RBF-QFs and Bayesian +quadrature. +Appendix A. Moments. +Henceforth, we provide the moments for different RBFs. The one-dimensional case is discussed in +§A.1, while two-dimensional moments are derived in §A.2. +A.1. One-dimensional moments. Let us consider the one-dimensional case of Ω = [a, b] and distinct +data points x1, . . . , xN ∈ [a, b]. + +TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS +25 +A.1.1. Gaussian RBF. For ϕ(r) = exp(−ε2r2), the moment of the translated Gaussian RBF, +(A.1) +mn = m(ε, xn, a, b) = +� b +a +exp(−ε2|x − xn|2) dx, +is given by +mn = +√π +2ε +� +erf(ε(b − xn)) − erf(ε(a − xn)) +� +. +Here, erf(x) = 2/√π +� x +0 exp(−t2) dt denotes the usual error function, [73, Section 7.2]. +A.1.2. Polyharmonic splines. For ϕ(r) = rk with odd k ∈ N, the moment of the translated PHS, +mn = m(xn, a, b) = +� b +a +ϕ(x − xn) dx, +is given by +mn = +1 +k + 1 +� +(a − xn)k+1 + (b − xn)k+1� +, +n = 1, 2, . . . , N. +For ϕ(r) = rk log r with even k ∈ N, on the other hand, we have +mn = (xn − a)k+1 +�log(xn − a) +k + 1 +− +1 +(k + 1)2 +� ++ (b − xn)k+1 +�log(b − xn) +k + 1 +− +1 +(k + 1)2 +� +. +Note that for xn = a the first term is zero, while for xn = b the second term is zero. +A.2. Two-dimensional moments. Here, we consider the two-dimensional case, where the domain +is given by a rectangular of the form Ω = [a, b] × [c, d]. +A.2.1. Gaussian RBF. For ϕ(r) = exp(−ε2r2), the two-dimensional moments can be written as +products of one-dimensional moments. In fact, we have +� b +a +� d +c +exp(−ε2∥(x − xn, y − yn∥2 +2) = m(ε, xn, a, b) · m(ε, yn, c, d). +Here, the multiplicands on the right-hand side are the one-dimensional moments from (A.1). +A.2.2. Polyharmonic splines and other RBFs. If it is not possible to trace the two-dimensional +moments back to the one-dimensional ones, we are in need of another approach. This is, for instance, +the case for PHS. We start by noting that for a data points (xn, yn) ∈ [a, b] × [c, d] the corresponding +moment can be rewritten as follows: +m(xn, yn) = +� b +a +� d +c +ϕ(∥(x − xn, y − yn)T ∥2) dy dx = +� ˜b +˜a +� ˜d +˜c +ϕ(∥(x, y)T ∥2) dy dx +with translated boundaries ˜a = a − xn, ˜b = b − xn, ˜c = c − yn, and ˜d = d − yn. We are not aware +of an explicit formula for such integrals for most popular RBFs readily available from the literature. +That said, such formulas were derived in [78, 80, 79] (also see [95, Chapter 2.3]) for the integral of ϕ +over a right triangle with vertices (0, 0)T , (α, 0)T , and (α, β)T . Assuming ˜a < 0 < ˜b and ˜c < 0 < ˜d, +we therefore partition the shifted domain ˜Ω = [˜a,˜b] × [˜c, ˜d] into eight right triangles. +Denoting the + +26 +J. GLAUBITZ AND J. A. REEGER +x +y +˜a +˜b +˜c +˜d +I1 +I2 +I3 +I4 +I5 +I6 +I7 +I8 +Figure 11: Illustration of how the moments can be computed on a rectangle in two dimensions +corresponding integrals by I1, . . . , I8, the moment m(xn, yn) correspond to the sum of these integrals. +The procedure is illustrated in Figure 11. +The special cases where one (or two) of the edges of the rectangle align with one of the axes can be +treated similarly. However, in this case, a smaller subset of the triangles is considered. We leave the +details to the reader, and note the following formula for the weights: +m(xn, yn) = +� +1 − δ0 +� +˜b ˜d +�� +(I1 + I2) + +� +1 − δ0 +� +˜a ˜d +�� +(I3 + I4) ++ +� +1 − δ0 (˜a˜c) +� +(I5 + I6) + +� +1 − δ0 +� +˜b˜c +�� +(I7 + I8) +Here, δ0 denotes the usual Kronecker delta defined as δ0(x) = 1 if x = 0 and δ0(x) = 0 if x ̸= 0. +The above formula holds for general ˜a, ˜b, ˜c, and ˜d. Note that all the right triangles can be rotated or +mirrored in a way that yields a corresponding integral of the form +(A.2) +Iref(α, β) = +� α +0 +� +β +αx +0 +ϕ(∥(x, y)T ∥2) dy dx. +More precisely, we have +I1 = Iref(˜b, ˜d), +I2 = Iref( ˜d,˜b), +I3 = Iref( ˜d, −˜a), +I4 = Iref(−˜a, ˜d), +I5 = Iref(−˜a, −˜c), +I6 = Iref(−˜c, −˜a), +I7 = Iref(−˜c,˜b), +I8 = Iref(˜b, −˜c). +Finally, explicit formulas of the reference integral Iref(α, β) over the right triangle with vertices (0, 0)T , +(α, 0)T , and (α, β)T for some PHS can be found in Table 3. Similar formulas are also available, for +instance, for Gaussian, multiquadric and inverse multiquadric RBFs. +We note that the approach presented above is similar to the one in [85], where the domain Ω = +[−1, 1]2 was considered. Later, the same authors extended their findings to simple polygons [84] using +the Gauss–Grenn theorem. Also see the recent work [86], addressing polygonal regions that may be +nonconvex or even multiply connected, and references therein. It would be of interest to see if these +approaches also carry over to computing products of RBFs corresponding to different centers or products +of RBFs and their partial derivatives, again corresponding to different centers. Such integrals occur as +elements of mass and stiffness matrices in numerical PDEs. In particular, they are desired to construct +linearly energy stable (global) RBF methods for hyperbolic conservation laws [35, 39, 40]. + +TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS +27 +ϕ(r) +Iref(α, β) +r2 log r +α +144 +� +24α3 arctan +� +β/α +� ++ 6β(3α2 + β2) log(α2 + β2) − 33α2β − 7β3� +r3 +α +40 +� +3α4 arcsinh +� +β/α +� ++ β(5α2 + 2β2) +� +α2 + β2 +� +r5 +α +336 +� +15α6 arcsinh +� +β/α +� ++ β(33α4 + 26α2β2 + 8β4) +� +α2 + β2 +� +r7 +α +3346 +� +105α8 arcsinh +� +β/α +� ++ β(279α6 + 326α4β2 + 200α2β4 + 48β6) +� +α2 + β2 +� +Table 3: The reference integral Iref(α, β)—see (A.2)—for some PHS +REFERENCES +[1] W. F. Ames, Numerical Methods for Partial Differential Equations, Academic Press, 2014. +[2] I. Aziz, W. Khan, et al., Numerical integration of multi-dimensional highly oscillatory, gentle oscillatory and +non-oscillatory integrands based on wavelets and radial basis functions, Engineering Analysis with Boundary +Elements, 36 (2012), pp. 1284–1295. +[3] I. 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Eins, Mathematische Annalen, 77 (1916), pp. 313–352. + diff --git a/i9FPT4oBgHgl3EQfFDQD/content/tmp_files/load_file.txt b/i9FPT4oBgHgl3EQfFDQD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..110075ae369f54c05705410d0200810aa3493295 --- /dev/null +++ b/i9FPT4oBgHgl3EQfFDQD/content/tmp_files/load_file.txt @@ -0,0 +1,1637 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf,len=1636 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='12998v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='NA] 30 Jan 2023 Towards stability results for global radial basis function based quadrature formulas∗ Jan Glaubitz† and Jonah Reeger‡ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Quadrature formulas (QFs) based on radial basis functions (RBFs) have become an essential tool for multivariate numerical integration of scattered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Although numerous works have been published on RBF-QFs, their stability theory can still be considered as underdeveloped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Here, we strive to pave the way towards a more mature stability theory for global and function-independent RBF-QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In particular, we prove stability of these for compactly supported RBFs under certain conditions on the shape parameter and the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' As an alternative to changing the shape parameter, we demonstrate how the least-squares approach can be used to construct stable RBF-QFs by allowing the number of data points used for numerical integration to be larger than the number of centers used to generate the RBF approximation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Moreover, it is shown that asymptotic stability of many global RBF-QFs is independent of polynomial terms, which are often included in RBF approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' While our findings provide some novel conditions for stability of global RBF-QFs, the present work also demonstrates that there are still many gaps to fill in future investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Numerical integration, radial basis functions, stability, cardinal functions, discrete orthogonal polynomials AMS subject classifications (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 65D30, 65D32, 65D05, 42C05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Numerical integration is an omnipresent task in mathematics and myriad appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' While these are too numerous to list fully, prominent examples include numerical differential equations [46, 76, 1], machine learning [68], finance [34], and biology [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In many cases, the problem can be formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let Ω ⊂ RD be a bounded domain with positive volume, |Ω| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Given N distinct data pairs {(xn, fn)}N n=1 ⊂ Ω×R with f : Ω → R and fn := f(xn), the aim is to approximate the weighted integral I[f] := � Ω f(x)ω(x) dx by an N-point QF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That is, by a weighted finite sum over the given function of the form CN[f] = N � n=1 wnf(xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In higher dimensions, CN is sometimes referred to as an N-point cubature formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The distinct points {xn}N n=1 are called data points and the {wn}N n=1 are referred to as quadrature weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Many QFs are derived based on the idea to approximate the (unknown) function f and then exactly integrate this approximation [43, 88, 25, 18, 57, 19, 58, 21, 9, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Arguably, most of the existing QFs have been derived from being exact for polynomials up to a certain degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' See [63, 69, 20, 70, 19, 93], in addition to the above references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That said, in recent years, QFs based on the exact integration of RBFs have received a growing amount of interest [87, 85, 84, 75, 2, 32, 78, 80, 95, 79, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The increased use of RBFs for numerical ∗January 31, 2023 Corresponding author: Jan Glaubitz (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='Glaubitz@Dartmouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='edu, orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='org/0000-0002-3434-5563) Disclaimer: The views expressed in this academic research paper are those of the authors and do not reflect the official policy or position of the United States Government or Department of Defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In accordance with the Air Force Instruction 51-303, it is not copyrighted, but is the property of the United States government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' †Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA ‡Air Force Institute of Technology, Wright–Patterson Air Force Base, OH 45433, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 1 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' GLAUBITZ AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' REEGER integration and numerical differential equations [55, 53, 26, 54, 60, 51, 83, 31, 28, 39, 40] seems to be only logical, considering their story of success in the last few decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In fact, since their introduction in Hardy’s work on cartography from 1971 (see [45]), RBFs have become a powerful tool in numerical analysis, including multivariate interpolation and approximation theory [12, 13, 98, 27, 52, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It should also be mentioned that RBF-QF can be connected to (statistical) Bayesian quadrature [72, 67, 10, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, recent literature in the quadrature area [78, 80, 79, 77] has focused on ’local’ RBF-FD-type implementations to reduce computational costs for large node numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' While an extension to such local approaches would be of interest, we restrict ourselves to global RBF methods in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That said, to reduce the cost of constructing and integrating a global interpolant, a piecewise RBF interpolant could be considered and integrated in a manner similar to the construction of Newton–Cotes formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Some of our results would easily carry over to this setting, which might be seen as an extreme version (no overlap of nonzero measure) of RBF partition of unity methods [3, 97, 27] or the overlapped RBF-FD methods [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Even though RBF-QFs have been proposed and applied in numerous works, their stability theory can still be considered as under-developed, especially compared to more traditional—e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' g polynomial based—methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Stability of RBF-QFs was broached, for instance, in [87, 85, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Further, stability of RBF-QF was discussed in [32] for integration on certain manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' However, to the best of our knowledge, an exhaustive stability theory for RBF-QFs is still missing in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In particular, theoretical results providing clear conditions under which stability of RBF-QFs is ensured are rarely encountered, even for global RBF methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The present work strives to fill this gap in the RBF literature partially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This is done by providing a detailed theoretical and numerical investigation on stability of global RBF-QFs1 for different families of kernels, including compactly supported and Gaussian RBFs as well as polyharmonic splines (PHS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Our analysis resembles classic stability theory for quadratures exact for polynomial spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In contrast to some existing works (see [15] and references therein), we consider RBF approximations with function- independent shape parameters to obtain quadrature formulas that do not have to be recomputed when another function is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In particular, we report on the following findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' (1) We provide a sufficient condition for compactly supported RBFs to yield a provable stable RBF-QF (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 in §4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The result is independent of the degree of the polynomial term that is included in the global RBF interpolant and assumes the data points to come from an equidistributed (space-filling) sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' (2) We demonstrate how the idea of least squares can be employed to construct provable stable RBF-QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' (3) Asymptotic stability of pure RBF-QFs is connected to asymptotic stability of the same RBF-QF but augmented with polynomials of a fixed arbitrary degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Essentially, we can show that for a sufficiently large number of data points, stability of RBF-QFs is independent of the presence of polynomials in the RBF interpolant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The rest of this work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We collect some preliminaries on RBF interpolants and QFs in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In §3, a few initial comments on stability of (RBF-)QFs are offered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Next, §4 contains our main theoretical result regarding stability of RBF-QFs based on compactly supported kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' §5 demonstrates how the concept of least squares can be used to construct provable stable RBF-QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Furthermore, it is proven in §6 that, under certain assumptions, asymptotic stability of RBF-QFs is independent of the polynomial terms included in the RBF interpolant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Numerical tests in §7 accompany the previous theoretical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, concluding thoughts are offered in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We collect some preliminaries on RBF interpolants (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) and RBF-QFs (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 1Henceforth, we will refer to these as “RBF-QFs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Radial basis function interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' RBFs are often considered a powerful tool in numerical analysis, including multivariate interpolation and approximation theory [12, 13, 98, 27, 52, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We are especially interested in RBF interpolants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let f : RD ⊃ Ω → R be a scalar valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Given a set of distinct data points (sometimes also referred to as centers), the RBF interpolant of f is of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) (sN,df)(x) = N � n=1 αnϕ(εn∥x − xn∥2) + K � k=1 βkpk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Here, ϕ : R+ 0 → R is the RBF (also called kernel), {pk}K k=1 is a basis of the space of algebraic polynomials up to degree d, Pd(Ω), and the εn’s are nonnegative shape parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 The RBF interpolant (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) is uniquely determined by the conditions (sN,df)(xn) = f(xn), n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) N � n=1 αnpk(xn) = 0, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3) Note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3) can be reformulated as a linear system for the coefficient vectors α = [α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , αN]T and β = [β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , βK]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This linear system is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4) � Φ P P T 0 � � α β � = � f 0 � , where f = [f(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , f(xN)]T as well as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5) Φ = \uf8ee \uf8ef\uf8ef\uf8f0 ϕ(ε1∥x1 − x1∥2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' ϕ(εN∥x1 − xN∥2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' ϕ(ε1∥xN − x1∥2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' ϕ(εN∥xN − xN∥2) \uf8f9 \uf8fa\uf8fa\uf8fb , P = \uf8ee \uf8ef\uf8ef\uf8f0 p1(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' pK(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' p1(xN) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' pK(xN) \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For a constant shape parameter ε1 = · · · = εN, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4) is ensured to have a unique solution—corresponding to existence and uniqueness of the RBF interpolant—if the kernel ϕ is conditionally positive definite of order d and the set of data points is Pd(Ω)-unisolvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' See, for instance, [27, Chapter 7] and [35, Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1] or references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In this work, we shall focus on the popular choices of RBFs listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A more complete list of RBFs and their properties can be found in the monographs [13, 98, 27, 30] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The set of all RBF interpolants (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) forms an N-dimensional linear space, denote by SN,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This space is spanned by the cardinal functions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6) cm(x) = N � n=1 α(m) n ϕ(εn∥x − xn∥2) + K � k=1 β(m) k pk(x), m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N, which are uniquely determined by the cardinal property (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7) cm(xn) = δmn := � 1 if m = n, 0 otherwise, m, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N, 2For polyharmonic splines, it is common practice to not include a shape parameter in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For simplicity, we still use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) and set εn = 1, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , n, in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' GLAUBITZ AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' REEGER RBF ϕ(r) parameter order Gaussian exp(−r2) 0 Wendland’s ϕD,k(r), see [96] D, k ∈ N0 0 Polyharmonic splines r2k−1 k ∈ N k r2k log r k ∈ N k + 1 Table 1: Some popular RBFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The “order” k of an RBF refers to the RBF being conditionally positive of order k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' and condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' They provide us with the following representation of the RBF interpolant (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1): (sN,df)(x) = N � n=1 f(xn)cn(x) This representation is convenient to subsequently derive quadrature weights based on RBFs that are independent of the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Quadrature formulas based on radial basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A fundamental idea behind many QFs is to first approximate the (unknown) function f : Ω → R based on the given data pairs {xn, fn}N n=1 ⊂ Ω×R and to exactly integrate this approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In the case of RBF-QFs this approximation is chosen as the RBF interpolant (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Hence, the corresponding RBF-QF is defined as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='8) CN[f] := I[sN,df] = � Ω (sN,df)(x)ω(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' When formulated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' the cardinal functions cn we get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='9) CN[f] = N � n=1 wnf(xn) with wn = I[cn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That is, the RBF quadrature weights w = [w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , wN]T are given by the moments corresponding to the cardinal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This formulation is often preferred over (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='8) since the weights w do not have to be recomputed when another function is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In our implementation, we compute the RBF quadrature weights by solving the linear system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='10) � Φ P P T 0 � � �� � =A � w v � = � mRBF mpoly � , where v ∈ RK is a Lagrange multiplier3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Furthermore, the vectors mRBF ∈ RN and mpoly ∈ RK re- spectively contain the moments of the translated kernels and polynomial basis functions: mRBF = � I[ϕ1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , I[ϕN] �T , mpoly = � I[p1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , I[pK] �T , 3The solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='10) can be interpreted as the solution of an equality constrained linear optimization problem [5], where v plays the role of a Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS 5 with ϕn(x) = ϕ(εn∥x − xn∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The moments of different RBFs can be found in the appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The polynomial moments can be found in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=', [37, Appendix A] and [29, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Stability and the Lebesgue constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This section addresses the stability of RBF interpolants and the corresponding RBF-QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In particular, we show that both can be estimated in terms of the Lebesgue constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This was also observed in [32] for RBF-QFs on certain (compact) manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That said, we also demonstrate that RBF-QFs often come with improved stability compared to RBF interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Stability of quadrature formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We shall start by addressing stability of RBF-QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' To this end, let us denote the best approximation of f from SN,d in the L∞-norm by ˆs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That is, ˆs = arg min s∈SN,d ∥f − s∥L∞(Ω) with ∥f − s∥L∞(Ω) = sup x∈Ω |f(x) − s(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Note that this best approximation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' the L∞-norm is not necessarily equal to the RBF interpolant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Still, the following error bound holds for the RBF-QF (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='9), that corresponds to exactly integrating the RBF interpolant from SN,d: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) |CN[f] − I[f]| ≤ � ∥I∥∞ + ∥CN∥∞ � inf s∈SN,d∥f − s∥L∞(Ω) Inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) is commonly known as the Lebesgue inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=', [94] or [9, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It is often encountered in polynomial interpolation [11, 49] but straightforwardly carries over to numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In this context, the operator norms ∥I∥∞ and ∥CN∥∞ are respectively given by ∥I∥∞ = I[1] and ∥CN∥∞ = N � n=1 |wn| = N � n=1 |I[cn]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Recall that the cn’s are the cardinal functions (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In fact, ∥CN∥∞ is a common stability measure for QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This is because the propagation of input errors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=', due to noise or rounding errors, can be bounded by ∥CN∥∞: Let ˜f : Ω → R be a perturbed version of f, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' including noise or measurement errors, then |CN[f] − CN[ ˜f]| ≤ ∥CN∥∞∥f − ˜f∥L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In other words, input errors are amplified at most by a factor that is equal to the operator norm ∥CN∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' At the same time, we have ∥CN∥∞ ≥ CN[1], where equality holds if and only if all quadrature weights are nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Also, for this reason, the construction of QFs is mainly devoted to nonnegative QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 (Stability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We call the RBF-QF CN stable if ∥CN∥∞ = CN[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This is the case if and only if I[cn] ≥ 0 for all cardinal functions cn, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It is also worth noting that CN[1] = ∥I∥∞ if the QF is exact for constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For RBF-QFs, this is the case if at least constants are included in the underlying RBF interpolant (d ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' GLAUBITZ AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' REEGER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Stability of RBF approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We now demonstrate how stability of the RBF-QF CN can be connected to stability of the corresponding RBF interpolant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Indeed, the stability measure ∥CN∥∞ can be bounded from above by ∥CN∥∞ ≤ ∥I∥∞ΛN, with ΛN := sup x∈Ω N � n=1 |cn(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Here, ΛN is the Lebesgue constant corresponding to the recovery process f �→ sN,df (RBF interpolation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Obviously, ΛN ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Also note that if 1 ∈ SN,d (the RBF-QF is exact for constants), we observe (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) ∥I∥∞ ≤ ∥CN∥∞ ≤ ∥I∥∞ΛN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Hence, the RBF-QF is stable (∥CN∥∞ = ∥I∥∞) if ΛN is minimal (ΛN = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We briefly note that the inequality ∥CN∥∞ ≤ ∥I∥∞ΛN is sharp by considering the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 (∥CN∥∞ = ΛN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let us consider the domain Ω = [0, 1] with ω ≡ 1, which immediately implies ∥I∥∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In [7] it was shown that for the linear PHS ϕ(r) = r and data points 0 = x1 < · · < xN = 1 the corresponding cardinal functions cm are simple hat functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In particular, cm is the ordinary “connect the dots” piecewise linear interpolant of the data pairs (xn, δnm), n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Thus, ΛN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' At the same time, this yields ∥CN∥∞ = 1 and therefore ∥CN∥∞ = ΛN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Looking for minimal Lebesgue constants is a classical problem in approximation and recovery theory [65, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For instance, it is well known that for polynomial interpolation, even near-optimal sets of data points yield a Lebesgue constant that grows as O(log N) in one dimension and as O(log2 N) in two dimensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' see [11, 6, 8, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In the case of RBF interpolation, the Lebesgue constant and appropriate data point distributions were studied, for instance, in [50, 22, 64, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That said, the second inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) also tells us that in some cases, we can expect the RBF-QF to have superior stability properties compared to the underlying RBF interpolant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, it should be stressed that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) only holds if 1 ∈ SN,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In general, we have CN[1] ≤ ∥CN∥∞ ≤ ∥I∥∞ΛN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Still, this indicates that a recovery space SN,d is desired that yields a small Lebesgue constant as well as the RBF-QF potentially having superior stability compared to RBF interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Compactly supported radial basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Despite the increased use of RBF-QFs in appli- cations, provable stability results are rarely encountered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' As a first step towards a more mature stability theory, we next prove stability of RBF-QFs for compactly supported kernels with nonoverlapping supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' To be more precise, we subsequently consider RBFs ϕ : R+ 0 → R satisfying the following restrictions: (R1) ϕ is nonnegative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=', ϕ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' (R2) ϕ is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' we assume maxr∈R+ 0 |ϕ(r)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' (R3) ϕ is compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' we assume supp ϕ = [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Already note that (R3) implies suppϕn = Bε−1 n (xn), where Bε−1 n (xn) := { x ∈ Ω | ∥xn − x∥2 ≤ ε−1 n }, ϕn(x) := ϕ(εn∥xn − x∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The ϕn’s will have nonoverlapping support if the shape parameters εn are sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This can be ensured by the following condition: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) ε−1 n ≤ hn := min � ∥xn − xm∥2 | xm ∈ X \\ {xn} � , n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS 7 Here, X denotes the set of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, it should be pointed out that throughout this section, we assume ω ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This assumption is made for the main result, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1, to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Its role will become clearer after consulting the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 and is revisited in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Our main result is the following Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' After collecting a few preliminary results, its proof is given in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let (xn)n∈N be an equidistributed sequence in Ω and XN = {xn}N n=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Furthermore, let ω ≡ 1, let ϕ : R+ 0 → R be a RBF satisfying (R1) to (R3), and choose the shape parameters εn such that the corresponding functions ϕn have nonoverlapping support and equal moments (I[ϕn] = I[ϕm] for all n, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For every polynomial degree d ∈ N there exists an N0 ∈ N such that for all N ≥ N0 the corresponding RBF-QF (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='9) is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That is, I[cm] ≥ 0 for all m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Note that a sequence (xn)n∈N is equidistributed in Ω if and only if lim N→∞ |Ω| N N � n=1 g(xn) = � Ω g(x) dx holds for all measurable bounded functions g : Ω → R that are continuous almost everywhere (in the sense of Lebesgue), see [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For details on equidistributed sequences, we refer to the monograph [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 Still, it should be noted that equidistributed sequences are dense sequences with a special ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In particular, if (xn)n∈N ⊂ Ω is equidistributed, then for every d ∈ N there exists an N0 ∈ N such that XN is Pd(Ω)-unisolvent for all N ≥ N0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' see [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This ensures that the corresponding RBF interpolant is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It should also be noted that if Ω ⊂ RD is bounded and has a boundary of measure zero (again in the sense of Lebesgue), then an equidistributed sequence in Ω is induced by an equidistributed sequence in the D-dimensional hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' More details on how an equidistributed sequence in Ω can be constructed are provided in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It is always possible to ensure the equal moment condition, I[ϕn] = I[ϕm] for all n, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N, in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 by allowing the points closer to the boundary to come with a smaller shape parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In this way, one can compensate for the part of the support cut off by the boundary of the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For instance, if equally spaced points are used on [a, b] with a = x1 < · · · < xN = b, then the shape parameter is εn = ε for the interior points (n = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N − 1) and ε1 = εN = ε/2 for the boundary points, where ε is a suitable chosen reference parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That said, in our numerical tests, we observed Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 also to hold when the equal moment condition was not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It is not necessary to include polynomials in the RBF-QF (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='9) for Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 to imply stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Indeed, it is subsequently proved by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 that the RBF-QF (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='9) can also be stable when no polynomials are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Sometimes, the RBF-QF (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='9) is also referred to as the “RBF+poly-QF” when polynomials are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In this regard, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 shows that stability of RBF-QFs carries over to RBF+poly-QFs under the assumptions listed in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The influence of including polynomials into the RBF-QFs on their stability is also discussed for other kernels in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Explicit representation of the cardinal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In preparation of proving Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 we derive an explicit representation for the cardinal functions cn under the restrictions (R1)–(R3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In particular, we use the concept of discrete orthogonal polynomials (DOPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let us define the 4Examples for equidistributed sequences include low-discrepancy points [47, 71, 14, 24] used in quasi-Monte Carlo methods, such as the Halton points [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' GLAUBITZ AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' REEGER following discrete inner product corresponding to the data points XN = {xn}N n=1: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) [u, v]XN = |Ω| N N � n=1 u(xn)v(xn) Recall that the data points XN are coming from an equidistributed sequence and are ensured to be Pd(Ω)-unisolvent for any degree d ∈ N if N is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In this case, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) is positive definite on Pd(Ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=', [u, u]XN > 0 if u ∈ Pd(Ω) and u ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We say that the basis {pk}K k=1 of Pd(Ω), where K = dim Pd(Ω), consists of DOPs if [pk, pl]XN = δkl := � 1 if k = l, 0 otherwise, k, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We now come to the desired explicit representation for the cardinal functions cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 (Explicit representation for cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let the RBF ϕ : R+ 0 → R satisfy (R2) and (R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Fur- thermore, choose the shape parameters εn such that the corresponding functions ϕn have nonoverlapping support and let the basis {pk}K k=1 consists of DOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Then, the cardinal function cm, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N, is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3) cm(x) = ϕm(x) − |Ω| N N � n=1 \uf8eb \uf8ed K � k=1 pk(xm)pk(xn) \uf8f6 \uf8f8 ϕn(x) + |Ω| N K � k=1 pk(xm)pk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let m, n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The restrictions (R2), (R3) together with the assumption of the ϕn’s having nonoverlapping support yields ϕn(xm) = δmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Hence, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7) imply (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4) α(m) n = δmn − K � k=1 β(m) k pk(xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' If we substitute (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3), we get pl(xm) − N |Ω| K � k=1 β(m) k [pk, pl]XN = 0, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Thus, if {pk}K k=1 consists of DOPs, this gives us (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5) β(m) l = N |Ω|pl(xm), l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4) yields α(m) n = δmn − N |Ω| K � k=1 pk(xm)pk(xn), and therefore the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It should be stressed that using a basis of DOPs is not necessary for implementing RBF-QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In fact, the quadrature weights are—ignoring computational considerations—independent of the polynomial basis w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' which the matrix P and the corresponding moments mpoly are formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We only use DOPs as a theoretical tool to show stability of RBF-QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Some low hanging fruits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Using the explicit representation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3) it is trivial to prove stability of RBF-QFs (I[cm] ≥ 0 for all m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N) when no polynomial term or only a constant is included in the RBF interpolant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 (No polynomials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let the RBF ϕ : R+ 0 → R satisfy (R1) to (R3) and choose the shape parameters εn such that the corresponding functions ϕn have nonoverlapping support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Assume that no polynomials are included in the corresponding RBF interpolant (K = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Then, the associated RBF-QF is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It is obvious that cm(x) = ϕm(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Thus, by restriction (R1), cm is nonnegative and therefore I[cm] ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6 (Only a constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let the RBF ϕ : R+ 0 → R satisfy (R1) to (R3) and choose the shape parameters εn such that the corresponding functions ϕn have nonoverlapping support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Assume that only a constant is included in the corresponding RBF interpolant (K = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Then, the associated RBF-QF is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' If we choose p1 ≡ |Ω|−1/2, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 yields cm(x) = ϕm(x) + 1 N \uf8eb \uf8ed1 − N � n=1 ϕn(x) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Note that by (R2), (R3), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1), we therefore have cm(x) ≥ ϕm(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Hence, (R1) implies the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Proof of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The following technical Lemma will be convenient to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let (xn)n∈N be equidistributed in Ω, XN = {xn}N n=1, and let [·, ·]XN be the discrete inner product (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Furthermore, let {p(N) k }K k=1 be a basis of Pd(Ω) consisting of DOPs w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' [·, ·]XN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Then, for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , K, p(N) k → pk in L∞(Ω), N → ∞, where {pk}K k=1 is a basis of Pd(Ω) consisting of continuous orthogonal polynomials satisfying � Ω pk(x)pl(x) dx = δkl, k, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Moreover, it holds that lim N→∞ � Ω p(N) k (x)p(N) l (x) dx = δkl, k, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The assertion is a combination of Lemma 11 and 12 from [37], where a general positive weight function ω was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Here, we only consider the case ω ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Essentially, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7 states that if a sequence of discrete inner products converges to a continuous one, then also the corresponding DOPs—assuming that the ordering of the elements does not change— converges to a basis of continuous orthogonal polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Furthermore, this convergence holds in a uniform sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We are now able to provide a proof for Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' GLAUBITZ AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' REEGER Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let d ∈ N and m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Under the assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1, we have I[ϕn] = I[ϕm] for all n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Thus, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4implies I[cm] = I[ϕm] \uf8eb \uf8ed1 − |Ω| N N � n=1 K � k=1 p(N) k (xm)p(N) k (xn) \uf8f6 \uf8f8 + |Ω| N K � k=1 p(N) k (xm)I[pk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let {p(N) k }K k=1 be a basis of Pd(Ω) consisting of DOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That is, [p(N) k , p(N) l ]XN = δkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In particular, p(N) 1 ≡ |Ω|−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' With this in mind, it is easy to verify that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6) |Ω| N N � n=1 K � k=1 p(N) k (xm)p(N) k (xn) = K � k=1 p(N) k (xm)|Ω|1/2[p(N) k , p(N) 1 ]XN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Thus, we have I[cm] ≥ 0 ⇐⇒ K � k=1 p(N) k (xm)I[p(N) k ] ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, observe that K � k=1 p(N) k (xm)I[p(N) k ] = |Ω|1/2 K � k=1 p(N) k (xm) � Ω p(N) k (x)p(N) 1 (x) dx, under the assumption that ω ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7 therefore implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7) lim N→∞ K � k=1 p(N) k (xm)I[p(N) k ] = 1, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='8 (On the assumption that ω ≡ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The assumption that ω ≡ 1 in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 is necessary for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7) to both hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' On the one hand, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6) is ensured by the the DOPs being orthogonal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' the discrete inner product (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This discrete inner product can be considered as an approximation to the continuous inner product ⟨u, v⟩ = � Ω u(x)v(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This also results in Lemma4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' On the other hand, in general, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7) only holds if the DOPs converge to a basis of polynomials that is orthogonal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' the weighted continuous inner product ⟨u, v⟩ω = � Ω u(x)v(x)ω(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Hence, for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7) to both hold true at the same time, we have to assume that ω ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In this case, the two continuous inner products are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Provable stable least squares RBF-QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 shows that compactly supported RBFs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Wendland’s kernels) can lead to stable interpolatory QFs if the shape parameter is so that none of the shifted kernels have a region of overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In our numerical tests, we observed this condition not just to be sufficient but also often being “close to” necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We often found the RBF-QF even to have negative weights when the support regions only slightly overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' At the same time, it is known that scaling Wendland’s kernels so that the support decreases with the number of data points results in the interpolation error to decrease only slowly or even to stagnate [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' To provide a more practical procedure for ensuring stability of RBF-QFs, we now demonstrate how a least-squares approach [48, 42, 36, 37] can be used to construct stable RBF-QFs by allowing the TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS 11 number of data points used for numerical integration to be larger than the number of centers that are used to generate the RBF approximation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The subsequent least-squares approach is not limited to compactly supported kernels and can be used to construct stable QFs that are exact for fairly general RBF approximation spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The only restrictions are that the RBF approximation space consists of continuous and bounded functions and contains constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Further, the number of data points used by quadrature has to be sufficiently larger than the dimension of the RBF approximation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Although the least-squares approach has recently been extended to general multi-dimensional function spaces that include constants in [38], the implications for RBF-QF have not yet been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' To this end, we consider a given center point set YM = {ym}M m=1, generating the M-dimensional RBF space SM,d, and a larger data point set XN = {xn}N n=1 with N > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Then, any QF CN[f] = �N n=1 wnf(xn) that is exact for all f ∈ SM,d has to satisfy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) \uf8ee \uf8ef\uf8ef\uf8f0 b1(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' b1(xN) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' bM(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' bM(xN) \uf8f9 \uf8fa\uf8fa\uf8fb � �� � =B \uf8ee \uf8ef\uf8ef\uf8f0 w1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' wN \uf8f9 \uf8fa\uf8fa\uf8fb � �� � =w = \uf8ee \uf8ef\uf8ef\uf8f0 I[b1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' I[bM] \uf8f9 \uf8fa\uf8fa\uf8fb � �� � =m , where {bm}M m=1 is a basis of SM,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The matrix B in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) depends on XN and YM (as well as on the kernel ϕ and the polynomial degree d), which we denote by B = B(XN, YM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Assume that the data point set XN is SM,d-unisolvent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=', f(xn) = 0, ∀xn ∈ XN =⇒ f ≡ 0 holds for all f ∈ SM,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 Then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) has infinitely many solutions, which form a (N − M)-dimensional affine linear subspace W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Every w ∈ W yields a QF that is exact for all functions from the RBF space SM,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We want to find a positive solution w ∈ W so that the corresponding QF with weights w is stable (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' To this end, we use the following result from [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6 in [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let Ω ⊂ RD, ω : Ω → R+ 0 be a Riemann integrable weight function that is positive almost everywhere, and let F = span{ bm | m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , M } be a finite-dimensional linear space of continuous and bounded functions that contains constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Further, let (xn)n∈N be an equidistributed sequence in Ω with ω(xn) > 0 for all n ∈ N and denote the affine linear subspace of solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) by WF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Then there exists an N0 ∈ N such that for all N ≥ N0 and discrete weights rn,N = |Ω|ω(xn)/N, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N, the corresponding least-squares QF CLS N [f] = N � n=1 wLS n f(xn) with wLS = arg min w∈WF ∥R−1/2w∥2, where R−1/2 = diag(1/√r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , 1/√rN), is positive and exact for all f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' If we apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 to the RBF function space SM,d, we get Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 5XN is SM,d-unisolvent, for instance, when the kernel ϕ is conditionally positive definite of order d and XN is Pd(Ω)- unisolvent, which is a common assumption to ensure uniqueness of RBF interpolants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' GLAUBITZ AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' REEGER Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let Ω ⊂ RD be compact and let ω : Ω → R+ 0 be a Riemann integrable weight function that is positive almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Further, let d ≥ 0 be an integer, let ϕ : R+ 0 :→ R be a continuous and conditionally positive kernel of order d, and let {ym}M m=1 be a given set of centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' If (xn)n∈N is an equidistributed sequence in Ω with ω(xn) > 0 for all n ∈ N, then there exists an N0 ∈ N such that for all N ≥ N0 and discrete weights rn,N = |Ω|ω(xn)/N, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N, the corresponding least-squares RBF-QF (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) CLS N [f] = N � n=1 wLS n f(xn) with wLS = arg min w∈W ∥R−1/2w∥2, where R−1/2 = diag(1/√r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , 1/√rN), is positive and exact for all f ∈ SM,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We first note that the RBF function space SM,d, which we defined in §2, is M-dimensional with M < ∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=', finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Because the kernel ϕ and all polynomials up to degree d are continuous, all functions from SM,d are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Further, since Ω ⊂ RD is compact and all functions from SM,d are continuous, they are also bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, d ≥ 0 implies that SM,d contains constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 now follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 with F = SM,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The weighted least-squares solution (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) has the advantage of being easy and efficient to compute using standard tools from linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The above discussion motivates us to formulate the following procedure to construct stable least-squares RBF-QFs (LSRBF-QFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 Constructing stable LSRBF-QFs 1: Input: Center points {ym}M m=1, kernel ϕ, polynomial degree d ≥ 0, weight function ω, and equidis- tributed data points (xn)n∈N 2: Output: An integer N ≥ M and a stable LSRBF-QF with points {xn}N n=1 and weights wLS ∈ RN 3: Set wLS equal to the weights of the interpolatory RBF-QF given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='10) 4: repeat 5: Increase the number of data points by one: N = N + 1 6: Set XN = {xn}N n=1 7: Compute the matrix B = B(XN, YM) as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) 8: Compute the weighted least-squares solution wLS as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) 9: Determine the smallest weight: wmin = min(wLS) 10: until wLS ≥ 0 Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 assumes that XM is SM,d-unisolvent, since the interpolatory RBF-QF given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='10) would not be defined otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The possible advantage of stable LSRBF-QFs compared to (potentially unstable) interpolatory RBF-QFs is demonstrated in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, we point out the potential application of stable LSRBF-QFs to the construction of stable RBF methods for time-dependent hyperbolic partial differential equations [90, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A crucial part of these methods is replacing exact integrals involving the approximate solution—in this case, an (local) RBF function— with a quadrature that should be as accurate as possible for functions from the approximation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Polynomial terms do not influence asymptotic stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Recall that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 in §4 holds regardless of the degree d of the polynomial term included in the RBF interpolant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Indeed, one might generally ask, “how are polynomial terms influencing stability of the RBF-QF?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In what follows, we TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS 13 address this question by showing that—under certain assumptions that are to be specified yet—at least asymptotic stability of RBF-QFs is independent of polynomial terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Recently, the following explicit formula for the cardinal functions was derived in [5, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let us denote c(x) = [c1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , cN(x)]T , where c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , cN are the cardinal functions spanning SN,d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Provided that Φ and P in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5) have full rank6, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) c(x) = ˆc(x) − Bτ(x) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Here, ˆc(x) = [ˆc1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , ˆcN(x)]T are the cardinal functions corresponding to the pure RBF interpolation without polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That is, they span SN,−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' At the same time, B and τ are defined as B := Φ−1P � P T Φ−1P �−1 , τ(x) := P T ˆc(x) − p(x) with p(x) = [p1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , pK(x)]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Note that τ can be interpreted as a residual measuring how well pure RBFs can approximate polynomials up to degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='9), we see that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) implies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) w = ˆw − BI[τ], where w is the vector of quadrature weights of the RBF-QF with polynomials (d ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' At the same time, ˆw is the vector of weights corresponding to the pure RBF-QF without polynomial augmentation (d = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Moreover, I[τ] denotes the componentwise application of the integral operator I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It was numerically demonstrated in [5] that for fixed d ∈ N one has (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3) max x∈Ω ∥Bτ(x)∥ℓ∞ → 0 as N → ∞ if PHS are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Note that, for fixed x ∈ Ω, Bτ(x) is an N-dimensional vector and ∥Bτ(x)∥ℓ∞ denotes its ℓ∞-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That is, the maximum absolute value of the N components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It should be pointed out that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3) was numerically demonstrated only for PHS in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' However, the relations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) hold for general RBFs as well as varying shape parameters, assuming that Φ and P have full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Please see [5, Section 4] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We also remark that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3) implies the weaker statement (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4) ∥Bτ(·)∥ℓ1 → 0 in L1(Ω) as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Here, Bτ(·) denotes a vector-valued function, Bτ : Ω → RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That is, for a fixed argument x ∈ Ω, Bτ(x) is an N-dimensional vector in RN and ∥Bτ(x)∥ℓ1 denotes the usual ℓ1-norm of this vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Thus, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4) means that the integral of the ℓ1-norm of the vector-valued function Bτ(·) converges to zero as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The above condition is not just weaker than (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3) (see Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4), but also more convenient to investigate stability of QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Indeed, we have the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let ω ∈ L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Assume Φ and P in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5) have full rank, and assume (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Then the two following statements are equivalent: (a) ∥ ˆw∥ℓ1 → ∥I∥∞ for N → ∞ (b) ∥w∥ℓ1 → ∥I∥∞ for N → ∞ That is, either both the pure and polynomial augmented RBF-QF are asymptotically stable or none is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A short discussion on the term “asymptotically stable” is subsequently provided in Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 6P having full rank means that P has full column rank, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=', the columns of P are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This is equivalent to the set of data points being Pd(Ω)-unisolvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' GLAUBITZ AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' REEGER Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Assume Φ and P in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5) have full rank, and assume (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) follows and therefore (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5) ∥w∥ℓ1 ≤ ∥ ˆw∥ℓ1 + ∥BI[τ]∥ℓ1, ∥ ˆw∥ℓ1 ≤ ∥w∥ℓ1 + ∥BI[τ]∥ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Next, note that BI[τ] = I[Bτ], and thus ∥BI[τ]∥ℓ1 = N � n=1 ��I[(Bτ)n] �� ≤ I \uf8ee \uf8f0 N � n=1 |(Bτ)n| \uf8f9 \uf8fb = I � ∥Bτ∥ℓ1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Since ω ∈ L∞(Ω), it follows that ∥BI[τ]∥ℓ1 ≤ ∥ω∥L∞(Ω) � Ω ∥Bτ(x)∥ℓ1 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Hence, by assuming that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4) holds, we get ∥BI[τ]∥ℓ1 → 0 for fixed d ∈ N and N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, substituting this into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5) yields the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 states that–under the listed assumptions—it is sufficient to consider asymptotic stabil- ity of the pure RBF-QF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Once asymptotic (in)stability is established for the pure RBF-QF, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1, it also carries over to all corresponding augmented RBF-QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Interestingly, this follows our find- ings for compactly supported RBFs reported in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' There, conditional stability was ensured independently of the degree of the augmented polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 (Asymptotic stability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We call a sequence of QFs with weights wN ∈ RN for N ∈ N asymptotically stable if ∥wN∥ℓ1 → ∥I∥∞ for N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Recall that ∥wN∥ℓ1 = ∥CN∥∞ if the weights wN correspond to the N-point QF CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It is easy to note that this is a weaker property than every single QF being stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=', ∥wN∥ℓ1 = ∥I∥∞ for all N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That said, consulting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1), asymptotic stability is sufficient for the QF to converge for all functions that can be approximated arbitrarily accurate by RBFs w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' the L∞(Ω)-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Of course, the propagation of input errors might be suboptimal for every single QF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 makes two assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' (1) Φ and P are full rank matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' and (2) the condition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In the two following remarks, we comment on these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3 (On the first assumption of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Although requiring A and P to have full rank might seem restrictive, there are often even more restrictive constraints in practical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For instance, when solving partial differential equations, the data points are usually required to be smoothly scattered so that the distance between data points is kept roughly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It seems unlikely to find A and P to be singular for such data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' See [5] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 (On the second assumption of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The second assumption for Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 to hold is that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That is, the integral of ∥Bτ(·)∥ℓ1 : Ω → R+ 0 converges to zero as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This is a weaker condition than the maximum value of ∥Bτ(·)∥ℓ1 converging to zero, which was numerically observed to hold for PHS in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The relation between these conditions can be observed by applying H¨older’s inequality (see, for instance, [81, Chapter 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let 1 ≤ p, q ≤ ∞ with 1/p+1/q = 1 and assume that ω ∈ Lq(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Then we have � Ω ∥Bτ(x)∥ℓ1ω(x) dx ≤ �� Ω ∥Bτ(x)∥p ℓ1 dx �1/p �� Ω ω(x)q dx �1/q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS 15 100 102 10-2 100 102 1/h (a) k = 1, d = −1 (pure RBF) 100 102 100 105 1/h (b) d = 0 (constant term) 100 102 100 105 1/h (c) d = 1 (linear term) Figure 1: The stability measure ∥CN∥∞ for Wendland’s compactly supported RBF ϕ1,k with smoothness param- eters k = 0, 1, 2 on N = 100 equidistant data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 1/h denotes the threshold above which the basis functions have nonoverlapping support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Hence, ∥Bτ∥ℓ1 converging to zero in Lp(Ω) as N → ∞ for some p ≥ 1 immediately implies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The special case of p = ∞ corresponds to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We present a variety of numerical tests in one and two dimensions to demon- strate our theoretical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A constant weight function ω ≡ 1 is used for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' All numerical tests presented here were generated in MATLAB7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Compactly supported RBFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let us start with demonstrating Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' To this end, we consider Wendland’s compactly supported RBFs in Ω = [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Figure 1 illustrates the stability measure ∥CN∥∞ of Wendland’s compactly supported RBF ϕ1,k with smoothness parameters k = 0, 1, 2 as well as the optimal stability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The latter is given by CN[1] if no constants are included and by ∥I∥∞ = 1 if constants are included in the RBF approximation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Furthermore, N = 100 equidistant data points were used, including the end points, x1 = 0 and xN = 1, and the (reference) shape parameter ε was allowed to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, 1/h denotes the threshold above which the compactly supported RBFs have nonoverlapping support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We note that the RBF-QFs are stable for sufficiently small shape parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' At the same time, we can also observe the RBF-QF be stable for ε ≥ 1/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It can be argued that this is in accordance with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Recall that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 essentially states that for ε ≥ 1/h, and assuming that all basis functions have equal moments (I[ϕn] = I[ϕm] for all n, m), the corresponding RBF-QF (including polynomials of any degree) is stable if a sufficiently large number of equidistribiuted data points is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Here, the equal moments condition was ensured by choosing the shape parameter as εn = ε for the interior data points (n = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N − 1) and as ε1 = εN = ε/2 for the boundary data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That said, at least numerically, we observe that it is possible to drop this equal moment condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This is demonstrated by Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' There, we perform the same test as in Figure 1, except choosing all the shape parameters to be equal (εn = ε, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N) and going over to nonequidistant Halton points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Nevertheless, we can see in Figure 2 that for ε ≥ 1/h the RBF-QFs are still stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Next, we extend our numerical tests to the following Genz test functions [33] (also see [94]) on 7See https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='com/jglaubitz/stability RBF CFs 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' GLAUBITZ AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' REEGER 100 102 100 101 1/h (a) d = 0 (constant term) 100 102 100 101 102 1/h (b) d = 1 (linear term) Figure 2: The stability measure ∥CN∥∞ for Wendland’s compactly supported RBF ϕ1,k with smoothness pa- rameters k = 0, 1, 2 on N = 100 Halton points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 1/h denotes the threshold above which the basis functions have nonoverlapping support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Ω = [0, 1]q: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) g1(x) = cos \uf8eb \uf8ed2πb1 + q � i=1 aixi \uf8f6 \uf8f8 , g2(x) = q � i=1 � a−2 i + (xi − bi)2�−1 , g3(x) = \uf8eb \uf8ed1 + q � i=1 aixi \uf8f6 \uf8f8 −(q+1) , g4(x) = exp \uf8eb \uf8ed− q � i=1 a2 i (xi − bi)2 \uf8f6 \uf8f8 Here, q denotes the dimension under consideration and is henceforth chosen as q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' These functions are designed to have different complex characteristics for numerical integration routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The vectors a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , aq)T and b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , bq)T respectively contain (randomly chosen) shape and translation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For each case, the experiment was repeated 100 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' At the same time, for each experi- ment, the vectors a and b were drawn randomly from [0, 1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For reasons of space, we only report the results for g1 and k = 1 in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' As before, the smallest errors are found for shape parameters cor- responding to the stable RBF-QF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The results for g2, g3, g4 and k = 0, 2 were similar and are therefore not reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Since it might be hard to identify the smallest errors as well as the corresponding shape parameter and stability measure from Figure 3, these are listed in Table 2 for d = 0, 1 together with the corresponding values for the fourth Genz test function g4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We see in Figure 3 that for d = −1 and increasing ε, the error increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This is because the supports of the translated kernels (disks in 2d) become smaller, resulting in “holes” in the pure RBF interpolant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=', regions where it is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In Figure 3c, for random points and d = −1, the supports become so small that all the quadrature weights become zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The holes vanish if at least a constant is included in the RBF interpolant, which explains the reduced errors for the same value of ε when d = 0 or d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, even for nonoverlapping supports (ε = 1/h), the area of the holes in the pure RBF part of the interpolant can converge to zero8 as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 8Assuming the sequence of points is dense in Ω TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS 17 10-1 101 103 10-6 10-2 102 1/h (a) Equidistant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' d = −1 10-1 101 103 10-6 10-2 102 1/h (b) Halton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' d = −1 10-1 101 103 10-6 10-2 102 1/h (c) Random,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' d = −1 10-1 101 103 10-6 10-2 102 1/h (d) Equidistant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' d = 0 10-1 101 103 10-6 10-2 102 1/h (e) Halton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' d = 0 10-1 101 103 10-6 10-2 102 1/h (f) Random,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' d = 0 10-1 101 103 10-6 10-2 102 1/h (g) Equidistant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' d = 1 10-1 101 103 10-6 10-2 102 1/h (h) Halton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' d = 1 10-1 101 103 10-6 10-2 102 1/h (i) Random,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' d = 1 Figure 3: Error analysis for Wendland’s compactly supported RBF ϕ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='k and the first Genz test function g1 on Ω = [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 1]2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' see (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In all cases, N = 400 data points (equidistant, Halton, or random) were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 1/h denotes the threshold above which the basis functions have nonoverlapping support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' If the RBF interpolant sN,df convergences to f in L1(Ω) as N → ∞, we get ��CN[f] − I[f] �� = ��I[sN,df] − I[f] �� ≤ � Ω |(sN,df)(x) − f(x)| dx → 0, N → ∞, and therefore CN[f] → I[f] as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For convergence results of RBF interpolants, we refer to the monographs [12, 98, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That said, we point out that the convergence of sN,df to f depends on the area not covered by the supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let us denote the area that is covered by the supports by Ωsupp(ϕ,XN,ε), then the area that is not covered by the supports is Ω\\Ωsupp(ϕ,XN,ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A rough but simple lower bound for the L1(Ω)-error of sN,df and f is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Note that the RBF interpolant is zero on Ω \\ Ωsupp(ϕ,XN,ε) 18 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' GLAUBITZ AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' REEGER g1 g4 emin ε ∥CN∥∞ emin ε ∥CN∥∞ Equidistant Points d = 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e+00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e+00 d = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e+00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e+00 Halton Points d = 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5e-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='9e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5e-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e+00 d = 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5e-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5e-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e+00 Random Points d = 0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5e-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6e-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='9e-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e+00 d = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2e-04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7e-04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0e+00 Table 2: Minimal errors, emin, for the first and fourth Genz test function, g1 and g4, together with the correspond- ing shape parameter, ε, and stability measure, ∥CN∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Wendland’s compactly supported RBF with smoothness parameter k = 1 was used in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' and thus (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) � Ω |(sN,df)(x) − f(x)| dx ≥ � Ω\\Ωsupp(ϕ,XN,ε) |f(x)| dx, where the right-hand side is the average absolute value of f in the area not covered by the supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) indicates that convergence includes the rate with which “holes” go to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In Figures 4 and 5, we relate the error in computing the integral of g1 on Ω = [0, 1]2 to the portion of the domain that is not covered by the possibly overlapping supports of the Wendland functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For equidistant points, the horizontal/vertical distance between adjacent points is (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' xij and x(i±1)j or xij and xi(j±1)) is 1/( √ N − 1), so as long as ε > 2( √ N − 1) the circles do not overlap and the total area that is not covered by the supports is given by 1 − π ε2 ( √ N − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Once ε ≤ 2( √ N −1), the circles overlap, and the total area that is not covered by the supports becomes 1 − π ε2 ( √ N − 1)2 + 2(θ − sin(θ)) ε2 ( √ N − 1)2, where θ = 2sin−1 \uf8eb \uf8ec \uf8ed � 4( √ N − 1)2 − ε2 2( √ N − 1) \uf8f6 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS 19 6 5 4 3 3 2 2 1 1 1 0 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 8 7 6 5 4 3 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='8 1 0 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='8 Figure 4: Error of CN[g1] on Ω = [0, 1]2 and the area not covered by the supports of the Wendland functions for various N and ε when no constant is included in the RBF interpolant 5 4 4 3 3 2 2 1 0 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 8 7 6 5 4 3 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='8 1 0 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='8 Figure 5: Error of CN[g1] on Ω = [0, 1]2 and the area not covered by the supports of the Wendland functions for various N and ε when a constant is included in the RBF interpolant Finally, when ε ≤ √ 2( √ N −1), the area that is not covered by the supports is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Figure 4 illustrates the error (left frame) and the area not covered by the supports (right frame) in this situation for various N and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The dashed lines represent the cases ε = 2( √ N − 1) and ε = √ 2( √ N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' On the other hand, Figure 5 illustrates the same test for an RBF interpolant that includes a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It demonstrates the improvement when the constant basis element covers the holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' GLAUBITZ AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' REEGER 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='8 1 (a) Random points 101 102 103 10-4 10-3 10-2 10-1 100 (b) Minimal and maximal distance Figure 6: Random points and the corresponding minimal and maximal distance, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The minimal distance is the smallest distance between any two distinct points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The maximal distance is the largest distance between any point and the closest distinct point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Stable LSRBF-QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We demonstrate that the least-squares approach discussed in §5 can stabilize RBF-QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We repeat that stable LSRBF-QF can be constructed for any RBF function space as long as we are willing to oversample, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=', the number of data points used by the quadrature is larger than the dimension of the RBF function space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In other words, there are more data points than center points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Notably, oversampling was used in some recent works [89, 90, 41] to stabilize RBF methods for partial differential equations, and it would be of interest to combine this with LSRBF-QF in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Here, we demonstrate the possible advantage of LSRBF-QFs compared to interpolatory RBF- QFs (data and center points are the same) for the RBF function space spanned by a constant and the functions ϕ(ε∥x − ym∥) on Ω = [0, 1]2 using a Gaussian kernel ϕ(r) = exp(−r2) and a constant (independent of the center and data points) shape parameter ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The center and data points, {ym}M m=1 and {xn}N n=1, are chosen as the first M and N elements of the same sequence of random points, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Figure 6a illustrates the first 64 random points from the sequence used in our tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Figure 6b visualizes the minimal and maximal distance of the random data point set XN = {xn}N n=1 for different values for N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We define the minimal distance of XN, denoted by hmin(XN), as the smallest distance between any two distinct points, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3) hmin(XN) = min xn∈XN min xm∈XN\\xn ∥xm − xn∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' At the same time, we define the maximal (filling) distance of XN, denoted by hmax(XN), as the largest distance between any point and the closest distinct point, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4) hmax(XN) = max xn∈XN min xm∈XN\\xn ∥xm − xn∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Figure 7a provides the values of the stability measure for the interpolatory RBF-QF (“RBF”) and the stable LSRBF-QF (“LSRBF”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For the same center points (same RBF function space for which the quadrature is exact), the LSRBF-QF uses more data points to evaluate the integrand than the TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS 21 101 102 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='5 (a) Stability measure 101 102 103 10-8 10-6 10-4 10-2 100 (b) Errors, no noise 101 102 103 10-6 10-4 10-2 100 (c) Errors, noise of magnitude 10−4 101 102 103 10-4 10-3 10-2 10-1 (d) Errors, noise of magnitude 10−2 Figure 7: Stability measure and errors for Genz’ first test function g1 on Ω = [0, 1]2 with ω ≡ 1 using an interpolatory RBF-QF and a stable LSRBF-QF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Random points and a Gaussian kernel with a constant shape parameter were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' interpolatory RBF-QF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The other way around, for the same data points, the interpolatory RBF-QF is exact for a larger RBF function space than the LSRBF-QF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' At the same time, the interpolatory RBF- QF is found to have a suboptimal stability measure (due to negative weights), which results in stability issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In contrast, in all cases, the LSRBF-QF has an optimal stability measure (due to the weights being positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Further, Figure 7b reports on the errors of the interpolatory RBF-QF and the stable LSRBF-QF on N random points applied to Genz’ first test function g1 on Ω = [0, 1]2 with ω ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In this example, both formulas perform similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In Figures 7c and 7d, we repeated this experiment but added uniformly distributed noise of magnitude 10−4 and 10−2 to the function values at the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The accuracy of the interpolatory RBF-QF deteriorates notably stronger than that of the LSRBF-QF in the presence of noise due to the improved stability of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We made the same observation also for other point distributions and Genz test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The LSRBF-QF having an optimal stability measure (being positive) for sufficiently large N can be explained by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 since we are given a compact domain, a positive weight function, and 22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' GLAUBITZ AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' REEGER 100 101 102 101 102 103 (a) Random points 100 101 102 101 102 103 (b) Halton points Figure 8: The smallest number of random/Halton data points, N, needed to find a positive LSRBF-QF that is exact for the RBF approximation space induced by the first M random/Halton center points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='8 1 (a) Halton points 101 102 103 10-2 10-1 100 (b) Minimal and maximal distance Figure 9: Halton points and the corresponding minimal and maximal distance, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The minimal distance is the smallest distance between any two distinct points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The maximal distance is the largest distance between any point and the closest distinct point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' a function space of continuous and bounded functions that contains constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Figure 8 reports the smallest number of random/Halton data points N we needed to find a positive LSRBF-QF that is exact for the RBF approximation space induced by using the first M random/Halton points as centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' To also illustrate the semi-random Halton points, Figure 9 visualizes the first 64 Halton points and their minimal and maximal distance for an increasing number N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Considering the model “N = C · Ms” and performing a least-squares fit for the parameters C and s given the data illustrated in Figure 8 revealed the following: For random data and center points, we found C ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1·10−1 and s ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For Halton data and center points, we found C ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='9·10−2 and s ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In both cases, we found N to be roughly linearly TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS 23 proportional to the squared dimension of the approximation space for which the positive least-squares quadrature is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Similar ratios were also observed in [48, 42, 36, 35, 37, 16, 17, 66, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It might be argued that the observed ratio between N and M is necessary for the LSRBF-QF to avoid inherent stability issues predicted by the ’impossibility’ theorem proved in [74], which states that any procedure for approximating univariate functions from equally spaced samples that converges exponentially fast must also be exponentially ill-conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, we address the convergence rate observed in Figure 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In theory, Gaussian RBF inter- polants can converge almost exponentially fast9 in the L∞(Ω) with the maximal (filling) distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For simplicity, we considered the model “|I[f] − CN[f]| = exp(−Chmax(XN)s)” and performing a least- squares fit for the parameters C and s using the data presented in Figure 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For the interpolatory RBF-QF, we found C ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='0 and s ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For the LSRBF-QF, we found C ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='8 and s ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Both QFs converge roughly exponentially, with the LSRBF-QF converging slightly faster than the interpola- tory RBF-QF, even in the noiseless case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A more general comment on the convergence of LSRBF-QFs is offered in Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2 (Convergence of LSRBF-QF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Assume that the positive LSRBF-QF CN on Ω is exact for all functions from the RBF approximation space SM,d(Ω) with d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Due to CN being positive and exact for constants, we have ∥CN∥∞ = ∥I∥∞ and the Lebesgue inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) implies |CN[f] − I[f]| ≤ 2∥I∥∞ � inf s∈SM,d(Ω)∥f − s∥L∞(Ω) � for any continuous f : Ω → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Now consider a sequence of positive LSRBF-QFs (CN)N∈N with CN being exact for SM,d(Ω) with M = M(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Assume that SM,d(Ω) ⊂ SM+1,d(Ω) for all M ∈ N and that the given function f lies in � M∈N SM,d(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Thus, if M(N) → ∞ for N → ∞, then (CN[f])N∈N converges to I[f] as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let us now assume that the ratio between M and N is of the form N = O(M2), which we numerically observed to be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The convergence rate of the LSRBF-QF for f is then the square root of the convergence rate of the best approximation of f from the sequence of approximation spaces (SM,d(Ω))N∈N for the L∞(Ω)-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Polyharmonic splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We end this section by providing a similar investigation for PHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Again, the first and fourth Genz test functions on Ω = [0, 1]2 are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' However, no shape parameter is involved for PHS, and we consider their stability and accuracy for an increasing number of Halton points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Figure 10 shows the results for the cubic (ϕ(r) = r3) and quintic (ϕ(r) = r5) PHS RBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We either added no polynomials (d = −1) or polynomial terms of order d = 0 and d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Figure 10 shows that all RBF-QFs converge while being stable or at least asymptotically stable, independent of the added polynomial term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In particular, we see that adding polynomial terms does not affect the asymptotic stability of the PHS RBF-QFs, per our results from §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, we see that the convergence rate of the RBF-QF depends on the kernel rather than being governed by the polynomial degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The added polynomial term reduces the error of the PHS RBF-QF but not the convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We observe second-order convergence for the cubic PHS RBF and third-order convergence for the quintic PHS RBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='10 9For a function from the appropriate native function space, the L∞(Ω)-error between the function and its RBF inter- polant is in O(exp(−C log hmax(XN)/hmax(XN)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' see [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 10In Figure 10a the cubic PHS RBF-QF first shows third-order convergence before it then settles for second-order convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We believe that the observed initial third-order decrease in the error is a combination of the second-order approximation rate of the cubic PHS-RBF interpolant and the decreasing Lebesgue constant ∥CN∥∞ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Once the QF is stable (∥CN∥∞ = ∥I∥∞), the second-order approximation rate dominates the error of the QF, and we thus start to observe second-order convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 24 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' GLAUBITZ AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' REEGER 101 102 103 10-10 100 (a) Cubic, d = −1 101 102 103 10-10 100 (b) Cubic, d = 0 101 102 103 10-10 100 (c) Cubic, d = 1 101 102 103 10-10 100 (d) Quintic, d = −1 101 102 103 10-10 10-5 100 (e) Quintic, d = 0 101 102 103 10-10 10-5 100 (f) Quintic, d = 1 Figure 10: Error analysis for the cubic (ϕ(r) = r3) and quintic (ϕ(r) = r5) PHS RBF in two dimensions using Halton points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The first and fourth Genz test functions g1, g4 were considered on Ω = [0, 1]2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' see (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Concluding thoughts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In this work, we investigated stability of RBF-QFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We started by show- ing that stability of RBF-QFs can be connected to the famous Lebesgue constant of the underlying RBF interpolant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This indicates that RBF-QFs might benefit from low Lebesgue constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Further- more, stability was proven for RBF-QFs based on compactly supported RBFs under the assumption of a sufficiently large number of (equidistributed) data points and the shape parameter(s) lying above a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, we showed that under certain conditions, asymptotic stability of RBF- QFs is independent of polynomial terms included in RBF approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A series of numerical tests accompanied the above findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' JG was supported by AFOSR #F9550-18-1-0316 and ONR #N00014-20-1- 2595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We thank Toni Karvonen for pointing out the connection between RBF-QFs and Bayesian quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Henceforth, we provide the moments for different RBFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The one-dimensional case is discussed in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1, while two-dimensional moments are derived in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' One-dimensional moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Let us consider the one-dimensional case of Ω = [a, b] and distinct data points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , xN ∈ [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS 25 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Gaussian RBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For ϕ(r) = exp(−ε2r2), the moment of the translated Gaussian RBF, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1) mn = m(ε, xn, a, b) = � b a exp(−ε2|x − xn|2) dx, is given by mn = √π 2ε � erf(ε(b − xn)) − erf(ε(a − xn)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Here, erf(x) = 2/√π � x 0 exp(−t2) dt denotes the usual error function, [73, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Polyharmonic splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For ϕ(r) = rk with odd k ∈ N, the moment of the translated PHS, mn = m(xn, a, b) = � b a ϕ(x − xn) dx, is given by mn = 1 k + 1 � (a − xn)k+1 + (b − xn)k+1� , n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For ϕ(r) = rk log r with even k ∈ N, on the other hand, we have mn = (xn − a)k+1 �log(xn − a) k + 1 − 1 (k + 1)2 � + (b − xn)k+1 �log(b − xn) k + 1 − 1 (k + 1)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Note that for xn = a the first term is zero, while for xn = b the second term is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Two-dimensional moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Here, we consider the two-dimensional case, where the domain is given by a rectangular of the form Ω = [a, b] × [c, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Gaussian RBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' For ϕ(r) = exp(−ε2r2), the two-dimensional moments can be written as products of one-dimensional moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In fact, we have � b a � d c exp(−ε2∥(x − xn, y − yn∥2 2) = m(ε, xn, a, b) · m(ε, yn, c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Here, the multiplicands on the right-hand side are the one-dimensional moments from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Polyharmonic splines and other RBFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' If it is not possible to trace the two-dimensional moments back to the one-dimensional ones, we are in need of another approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' This is, for instance, the case for PHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We start by noting that for a data points (xn, yn) ∈ [a, b] × [c, d] the corresponding moment can be rewritten as follows: m(xn, yn) = � b a � d c ϕ(∥(x − xn, y − yn)T ∥2) dy dx = � ˜b ˜a � ˜d ˜c ϕ(∥(x, y)T ∥2) dy dx with translated boundaries ˜a = a − xn, ˜b = b − xn, ˜c = c − yn, and ˜d = d − yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We are not aware of an explicit formula for such integrals for most popular RBFs readily available from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' That said, such formulas were derived in [78, 80, 79] (also see [95, Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='3]) for the integral of ϕ over a right triangle with vertices (0, 0)T , (α, 0)T , and (α, β)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Assuming ˜a < 0 < ˜b and ˜c < 0 < ˜d, we therefore partition the shifted domain ˜Ω = [˜a,˜b] × [˜c, ˜d] into eight right triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Denoting the 26 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' GLAUBITZ AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' REEGER x y ˜a ˜b ˜c ˜d I1 I2 I3 I4 I5 I6 I7 I8 Figure 11: Illustration of how the moments can be computed on a rectangle in two dimensions corresponding integrals by I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' , I8, the moment m(xn, yn) correspond to the sum of these integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The procedure is illustrated in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The special cases where one (or two) of the edges of the rectangle align with one of the axes can be treated similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' However, in this case, a smaller subset of the triangles is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We leave the details to the reader, and note the following formula for the weights: m(xn, yn) = � 1 − δ0 � ˜b ˜d �� (I1 + I2) + � 1 − δ0 � ˜a ˜d �� (I3 + I4) + � 1 − δ0 (˜a˜c) � (I5 + I6) + � 1 − δ0 � ˜b˜c �� (I7 + I8) Here, δ0 denotes the usual Kronecker delta defined as δ0(x) = 1 if x = 0 and δ0(x) = 0 if x ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' The above formula holds for general ˜a, ˜b, ˜c, and ˜d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Note that all the right triangles can be rotated or mirrored in a way that yields a corresponding integral of the form (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content='2) Iref(α, β) = � α 0 � β αx 0 ϕ(∥(x, y)T ∥2) dy dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' More precisely, we have I1 = Iref(˜b, ˜d), I2 = Iref( ˜d,˜b), I3 = Iref( ˜d, −˜a), I4 = Iref(−˜a, ˜d), I5 = Iref(−˜a, −˜c), I6 = Iref(−˜c, −˜a), I7 = Iref(−˜c,˜b), I8 = Iref(˜b, −˜c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Finally, explicit formulas of the reference integral Iref(α, β) over the right triangle with vertices (0, 0)T , (α, 0)T , and (α, β)T for some PHS can be found in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Similar formulas are also available, for instance, for Gaussian, multiquadric and inverse multiquadric RBFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' We note that the approach presented above is similar to the one in [85], where the domain Ω = [−1, 1]2 was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Later, the same authors extended their findings to simple polygons [84] using the Gauss–Grenn theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Also see the recent work [86], addressing polygonal regions that may be nonconvex or even multiply connected, and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' It would be of interest to see if these approaches also carry over to computing products of RBFs corresponding to different centers or products of RBFs and their partial derivatives, again corresponding to different centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' Such integrals occur as elements of mass and stiffness matrices in numerical PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' In particular, they are desired to construct linearly energy stable (global) RBF methods for hyperbolic conservation laws [35, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} +page_content=' TOWARDS STABILITY RESULTS FOR GLOBAL RBF-QFS 27 ϕ(r) Iref(α, β) r2 log r α 144 � 24α3 arctan � β/α � + 6β(3α2 + β2) log(α2 + β2) − 33α2β − 7β3� r3 α 40 � 3α4 arcsinh � β/α � + β(5α2 + 2β2) � α2 + β2 � r5 α 336 � 15α6 arcsinh � β/α � + β(33α4 + 26α2β2 + 8β4) � α2 + β2 � r7 α 3346 � 105α8 arcsinh � β/α � + β(279α6 + 326α4β2 + 200α2β4 + 48β6) � α2 + β2 � Table 3: The reference integral Iref(α, β)—see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FPT4oBgHgl3EQfFDQD/content/2301.12998v1.pdf'} 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+Barnard College, Columbia University +bplancher@barnard.edu, USA +Vijay Janapa Reddi +Harvard University +vj@eecs.harvard.edu, USA +Abstract +The sustained growth of carbon emissions and global waste +elicits significant sustainability concerns for our environ- +ment’s future. The growing Internet of Things (IoT) has the +potential to exacerbate this issue. However, an emerging area +known as Tiny Machine Learning (TinyML) has the opportu- +nity to help address these environmental challenges through +sustainable computing practices. TinyML, the deployment of +machine learning (ML) algorithms onto low-cost, low-power +microcontroller systems, enables on-device sensor analyt- +ics that unlocks numerous always-on ML applications. This +article discusses the potential of these TinyML applications +to address critical sustainability challenges. Moreover, the +footprint of this emerging technology is assessed through a +complete life cycle analysis of TinyML systems. From this +analysis, TinyML presents opportunities to offset its carbon +emissions by enabling applications that reduce the emis- +sions of other sectors. Nevertheless, when globally scaled, +the carbon footprint of TinyML systems is not negligible, +necessitating that designers factor in environmental impact +when formulating new devices. Finally, research directions +for enabling further opportunities for TinyML to contribute +to a sustainable future are outlined. +1 +Introduction +The continued growth of carbon emissions and global waste +presents a great concern for our environment, increasing +calls for a more sustainable future. In response, the United +Nations’ (UN) 2030 Agenda for Sustainable Development +established a shared framework aiming toward peace and +prosperity for people and the planet. At its core are 17 Sus- +tainable Development Goals (SDGs) [36], a call to action +for all countries to work towards a more environmentally, +economically, and socially sustainable future. +Tiny machine learning (TinyML), i.e. the ability to run ML +on microcontroller devices, unlocks an expansive new array +of always-on ML applications that can help address many of +Maintain +Design +Deploy +TinyML +Cycle +Manufacturing +Energy +Raw +Materials +Microcontrollers +Logistics Costs +End-of-life +Recycling +Upcycling +E-waste +Energy Consumption +Sustainable +Development Goals +Figure 1. TinyML can contribute to the UN’s environmen- +tal sustainability goals. However, the life cycle of TinyML +systems must be considered to maximize their global ben- +efit. The positive (green arrows) and negative (red arrows) +environmental footprint of TinyML systems are shown. +the UN’s SDGs, especially those that target environmental +sustainability (see Figure 1). Often, only the operational ben- +efits of TinyML are noted for environmental sustainability. +However, in this article we argue that the entire life cycle of +both TinyML applications and hardware must be considered +to ensure that these new technologies provide more carbon +savings than emissions. To this end, the contributions of the +paper are the following: (1) an assortment of case studies +demonstrating TinyML’s sustainability benefits is provided, +(2) the environmental impacts of TinyML are outlined at +the individual MCU-level and at the complete, integrated +TinyML system-level through a life cycle analysis, and (3) +future work is identified in enabling sustainable TinyML. +The rest of the paper is structured as follows. Section 2 +provides background on TinyML. Section 3 explores sus- +tainable applications enabled by TinyML. Section 4 uses a +complete life cycle analysis to assess the environmental foot- +print of a single MCU and complete TinyML system along +arXiv:2301.11899v1 [cs.LG] 27 Jan 2023 + +PROTECTRESTOREANDF +PROMOTESUSTAINABLEUSEOFTERRESTRIAI +ECOSYSTEMSSUSTAINABLYMANAGEFORESTS,COMBATDESERTIFICATION +AND HALTANDREVERSELAND DEGRADATIONAND HALT BIODIVERSITYLOSS +MORETHAN AOUARTER OESPEHIES +PROGRESSTOSAFEGUARD +KEY BIODIVERSITYAREASHAS +SEDBYTHEUCNRED +AR +STALLED OVERTHELAST5YEARS +HREATENED WITHEXTINCTIOR +GLOBAL MEAN PERCENTAGE +OF EACH KEY BIODIVERSITY AREA +PROPORTION OF SPECIES THREATENED WITH EXTINCTION +COVERED BY PROTECTED AREAS(2021) +信43% +TERRESTRIAL +42% +FRESHWATER +14% +41% +41% +34% +33% +26% +MOUNTAIN +AMPHIBIANS +CONIFERS +REEF-BUILDING +MAMMALS +BIRDS +CORALS +ALMOSTALLCOUNTRIESHAVEADOPTED +LEGISLATIONFORPREVENTING +JUCNREDLIST +TRACKSDATAONMORETHAN134,400SPECIESOFMAMMALS, +OR +BIRDS.AMPHIBIANS.REEF-BUILDING CORALS AND CONIFERS +CONTROLLINGINVASIVEALIEN SPECIES +MORETHAN37.400SPECIESARE THREATENEDWITHEXTINCTION +PROGRESS HASBEEN MADE +ARIA +SUSTAINABLE FOREST MANAGEMENT +BUTTHEWORLDHAS LOST +100 MILLIONHECTARESOFFOREST +INTWODECADES +[2000-2020] +INVASIVEALIEN SPECIESNEGATIVELY AFFECTNATIVE BIODIVERSITY +AND COSTTHE GLOBAL ECONOMY BILLIONS OF DOLLARS ANNUALLY +THESUSTAINABLEDEVELOPMENTG0ALSREPORT2021:UNSTATS.UN.0RG/SDGS/REPORT/2021/3 +GOODHEALTH +ANDWELL-BEING +ENSUREHEALTHYLIVESAND PROMOTE +M +WELL-BEING FOR ALLAT ALLAGES +COVID-19 +ISTHREATENINGDECADESOFPROGRESSINGLOBALHEALTH +INFECTED MORE THAN +LED TO +DISRUPTED ESSENTIAL +HALTED +15MILLION介 +HEALTH SERVICESIN +PROGRESS ON +500MILLION +PEOPLE +DEATHS介 +92% OF +UNIVERSAL +COUNTRIES +HEALTH +WORLDWIDE +COVERAGE +[MID-2022] +[2020-2021] +END2021] +GLOBAL +Emm +PREVALENCE OF +DEATHSFROM +LIFE +IMMUNIZATION +ANXIETY / +TUBERCULOSIS +EXPECTANCY +COVERAGE +DEPRESSION +GMALARIA +22.7 MILLI0N +CHILDREN +M +PANDEMIC CLAIMED THE LIVES OF +MISSED BASIC +115,500 FR0NT-LINE +VACCINESIN2020 +HEALTH-CARE WORKERS +0 +M0RETHANIN2019 +TUBERCULOSISDEATHS +MILLION +MILLION +RISEF0RTHEFIRSTTIMESINCE2005 +2019 +2020ZERO +HUNGER +END HUNGER-ACHIEVEFOOD SECURITY AND IMPROVEI +SSS +NUTRITION ANDPROMOTE SUSTAINABLE AGRICULTURE +THEGLOBALPANDEMIC +PANDEMIC WILL WORSEN +ISEXACERBATING +22% +[149.2MILLION] +WORLD HUNGER +OF CHILDRENUNDER5 +ARE STUNTED +6.7% (45.4 MILLION) +WORLDWIDE.ANADDITIONAL +OF CHILDRENUNDER5 +70-161MILLIONPE0PLE +SUFFER FROM WASTING +ARELIKELY TO HAVE +5.7% +(38.9MILLION) +EXPERIENCED HUNGER +ASARESULTOFTHE +OF CHILDREN UNDER5 +ARE0VERWEIGHT[2020*) +PANDEMICIN2020 ++THESE2020ESTIMATESDONOTREFLECTIMPACTOFPANDEMIC +ALMOST +NUMBEROFUNDERNOURISHED PEOPLEIN THE WORLD +ONETHIRD OF WOMEN +OF REPRODUCTIVEAGE +607 +650 +720-811 +GLOBALLYSUFFER FROM +MILLION +MILLION +MILLION +ANAEMIA.JIN PART DUETO +NUTRITION DEFICIENCIES +2014 +2019 +2020 +2.37 BILLION PEOPLEAREWITHOUT FO0D +ORUNABLETO EATA HEALTHY BALANCED DIET +ON A REGULAR BASIS(2020) +THESUSTAINABLEDEVELOPMENTGOALSREPORT2021:UNSTATS.UN.ORG/SDGS/REPORT/2021/CLEANWATER +AND SANITATION +ENSURE AVAILABILITY AND SUSTAINABLE +MANAGEMENTOFWATERAND SANITATIONFOR ALL +BILLIONSOEPEOPLESTILLLACK +ACCESS TO SAFEDRINKINGWATER +736777 +SANITATION AND HYGIENE +2.3BILLION PEOPLE +IN 2020 +LIVEIN +WATER-STRESSED +COUNTRIES +[2018] +2BILLION PEOPLE +3.6BILLIONPEOPLE +2.3BILLIONPEOPLE +26% +46% +29% +LACK +LACK +LACK +BETWEEN1970AND2015 +SAFELY MANAGED +SAFELY MANAGED +BASIC +DRINKING WATER +SANITATION +HYGIENE +NATURALWETLANDS +SHRANK BY 35% +ENSURING UNIVERSALACCESS IS FUNDAMENTAL +FOR COVID-19 RECOVERY +3XTHERATEOFFORESTLOSS +129COUNTRIESARENOTONTRACKTOHAVE +SUSTAINABLYMANAGEDWATERRESOURCESBY2O30 +RATE OEPROGRESS NEEDS TO DOUBLE +THESUSTAINABLEDEVEL0PMENTG0ALSREPORT2021:UNSTATS.UN.0RG/SDGS/REPORT/2021/12 +RESPONSIBLE +CONSUMPTION +ENSURESUSTAINABLECONSUMPTION +AND PRODUCTION +8 +AND PRODUCTION PATTERNS +THEGLOBAL“MATERIALFOOTPRINT +ELECTRONICWASTE +CONTINUES TO PROLIFERATE +NCREASEDBY70% +AND IS NOTDISPOSED OFRESPONSIBLY +BETWEEN 2000 AND2017 +EACH PERSON +GENERATED ABOUT +7.3KIL0GRAMS +BUT ONLY +OFE-WASTE +1.7 KILOGRAMS +2000 +WAS RECYCLED +[2019] +1MILLION +5TRILLION +PLASTIC DRINKING BOTTLES +SINGLE-USEPLASTICBAGS +AREPURCHASED +ARETHROWNAWAY +DESPITEPROGRESS +EVERY MINUTE +EACHYEAR +FOSSILFUEL SUBSIDIES CONTINUE +TO THREATEN THEACHIEVEMENTOF +DEVELOPING COUNTRIES +THEPARIS AGREEMENT AND +STILLHAVEVASTUNTAPPEDPOTENTIAL +2030AGENDA +FOR RENEWABLE ENERGY +NEW RENEWABLE ELECTRICITYCAPACITY +S432BILLI0NIN2019 +A DECLINEOF21%FROM 2018 +880 WATTSPERCAPITA +219WATTS PER CAPITA +4X +DEVELOPED COUNTRIES +DEVELOPING COUNTRIES +BY2020 +ATOTALOF7O0POLICIES +UNDERTHE1O-YEARFRAMEWORKOFPROGRAMMES +ANDIMPLEMENTATIONACTIVITIES +ON SUSTAINABLI +H +WERE REPORTED +(FROM83COUNTRIESANDTHEEUROPEANUNION)TAKE URGENT ACTION TO COMBAT +CLIMATE CHANGE AND ITS IMPACTS +RISING +THECLIMATECRISIS +GREENHOUSE GASEMISSIONS +CONTINUES +REQUIRESHIFTING ECONOMIES +TOWARDS CARBON NEUTRALITY +LARGELYUNABATED +CURRENT GREENHOUSE +GAS EMISSIONS +1.5°CSCENARI0 +2000 +2019 +2050 +CLIMATEFINANCE +INCREASED +2020 GLOBALAVERAGE TEMPERATUREAT +BY10% +FROM 2015-2016 +TO2017-2018 +REACHING AN +ANNUALAVERAGEOF +TRACKTO ST +S48.7BILLION +HIGHEST PRIORITY AREAS INCLUDE +1250F154DEVEL0PINGCOUNTRIES +AREFORMULATING ANDIMPLEMENTING +FOOD +TERRESTRIAL +FRESHWATER +HUMAN +KEY ECONOMIC +NATIONALCLIMATEADAPTATIONPLANS +SECURITY AND +ANDWETLAND +RESOURCES +HEALTH +SECTORS AND +PRODUCTION +ECOSYSTEMS +SERVICESCONSERVE AND SUSTAINABLY USE THE OCEANS.SEA AND +MARINERESOURCESFORSUSTAINABLEDEVELOPMENT +THE SUSTAINABILITY +DEAD ZONES +ARERISINGATANALARMINGRATE +OF OUR OCEANSIS +FR0M400IN2008T0700IN2019 +UNDERSEVERETHREAT +PLASTIC/MARINE POLLUTION +美 +几个 +DEADZONES"AREAREASOFWATERTHATLACK +SUFFICIENT OXYGEN TO SUPPORT MARINELIFE +FISHERY +OCEAN +OVERHALFOF +COLLAPSE +WARMING +MARINEKEY BIODIVERSITYAREAS +ARENOTPROTECTED +CO +ACIDIFICATION +EUTROPHICATION +OVER3BILLIONPEOPLE +RELY ON OCEANSFORTHEIRLIVELIHOODS +ABOUTHALFOF COUNTRIESWORLDWIDE +ON AVERAGE.0NLY 1.2% +HAVEADOPTED SPECIFICINITIATIVES +OFNATIONALRESEARCHBUDGETSARE +TO SUPPORT SMALL-SCALEFISHERS +ALLOCATED FOR OCEAN SCIENCE +333333 +5333333 +THESUSTAINABLEDEVELOPMENTG0ALSREPORT2021:UNSTATS.UN.0RG/SDGS/REPORT/2021/with the impact at scale. Finally, future research directions +for Sustainable TinyML are discussed in Section 5. +2 +Tiny Machine Learning (TinyML) +TinyML is the deployment of machine learning (ML) algo- +rithms onto low-cost, low-power, and resource-constrained +MCU systems. TinyML stores neural network models directly +within memory (e.g., flash) and runs inference directly on the +output of onboard sensors. This approach enables intelligent +on-device sensor analytics unavailable with traditional Inter- +net of Things (IoT) approaches, which instead typically rely +on communication with the cloud to transmit data for exter- +nal processing. Importantly, TinyML achieves this using a +fraction of the compute resources needed for traditional ML +systems. Table 1 compares TinyML with traditional BigML +(such as cloud and mobile systems) and shows how TinyML +requires orders of magnitude fewer resources across com- +pute, memory, storage, power, and cost. Finally, while the +heterogeneity and limited resources of MCU devices present +new challenges for on-device training, model updating, and +deployment, recent research and the development of ML +frameworks such as TensorFlow Lite for Microcontrollers [8] +have increased the accessibility of TinyML. +The ubiquity, low-cost, and small power envelope of MCUs, +paired with TinyML’s independence from internet connec- +tivity, enables ML models to be deployed globally anywhere +at scale. For these reasons, along with bandwidth, latency, +energy, reliability, and privacy concerns, running ML directly +on these embedded edge devices is growing in popularity. +With more than 250 billion MCUs deployed globally today, +and the cost of MCUs expected to drop below $0.50 per unit, +this number is expected to grow, eclipsing 50 billion MCUs +shipped per annum in the next decade [31]. As such, TinyML +will become an ever-present technology. But the question +we must ask ourselves is do we run the risk of producing an +Internet of Trash over the course of TinyML devices’ lifetime? +3 +Applications of TinyML for +Sustainability +To fairly evaluate the environmental impacts of machine +learning on microcontrollers, we first consider TinyML’s ben- +efits, given that the creation of value is what will likely result +in widespread TinyML deployments. Typical well-known +consumer-facing applications of TinyML include keyword +spotting, image classification, and anomaly detection [3]. +However, many other applications of TinyML can be used +to enable a more sustainable future [2]. In the following sec- +tions, emerging applications are highlighted which show +how TinyML can help make progress towards important +environmental-related SDGs (as shown in Figure 1). In par- +ticular, TinyML is shown to be well-suited for improving the +sustainability of global agriculture, aiding wildlife conserva- +tion, and helping combat climate change and its impacts. +Platform +Freq. +Memory +Storage +Power +Price +CO2-eq Footprint +Cloud +GHz +10+GB +TBs-PBs +∼1 kW +$1000+ +Hundreds of kgs +Mobile +GHz +Few GB +GBs +∼1 W +$100+ +Tens of kgs +Tiny +MHz +KBs +Few MB +∼1 mW +$10 +Single kgs +Table 1. Cloud and mobile ML systems are compared to +TinyML systems (in bold) across frequency, memory, store, +power, price, and footprint. This work shows the footprint +of TinyML systems is on the order of a few kilograms of +CO2-eq, far less than cloud and mobile ML systems. +3.1 +Zero Hunger & Good Health and Well-Being +(SDG #2 & #3) +End hunger, achieve food security & improved nutrition, pro- +mote sustainable agriculture, and ensure healthy lives while +promoting well-being for all at all ages. +ML applications can increase agriculture production through +data-driven methods. For example, Nuru, a mobile and cloud- +based ML app from the PlantVillage project, is more accu- +rate than humans at detecting plant diseases and enabled +one farmer to increase her revenue by 55% and yields by +146% [6, 24]. ML has also been used for autonomous devices +such as tiny drones, which can provide targeted pesticide +applications that reduce use to 0.1% of conventional blanket +spraying [21]. As another example, researchers developed a +cough monitor system to flag respiratory problems in pigs +by placing microphones over animal pens which can alert +farmers 12 days earlier than standard methods [21]. +TinyML has the potential to increase the impact of these +systems. First, it can enable these and many other appli- +cations to be used in remote regions through low-power, +low-connectivity operations. Second, it can enable scaled +deployment of these smart sensors, which could provide +more targeted information (e.g., on all individual pigs in real- +time for the cough monitor system). Most importantly, it +would increase global access to these technologies by reduc- +ing costs. As Sparrow and Howard note, global adoption can +only occur if devices “can be manufactured and sold cheaply +enough to be available to smaller farms” [30]. +On another very important and serious note, TinyML can +also be used to aid in our health and well-being. One of the +diseases noted in the UN’s SDG report when considering our +well-being is malaria due to its massive global impact that +spans a long history. In fact, nearly half of the whole world +population has been killed by mosquitoes [38]. Gaps in fund- +ing and access to life-saving tools led to a disproportionate +94% of all malaria cases and deaths in 2019 occurring in the +African region [36]. Using Edge Impulse, a development plat- +form for TinyML, a system was prototyped to identify the +deadliest mosquitoes using wing beats sound classification +with 88.3% accuracy [35]. This is another example in which +global access to these systems will have a tremendous impact +and could potentially save lives. +2 + +3.2 +Life on Land & Below Water (SDGs #14 & #15) +Protect, restore, conserve, and promote sustainable use of ter- +restrial & aquatic ecosystems, sustainably manage forests & +marine resources, combat desertification, halt biodiversity loss. +TinyML can help preserve the planet’s biodiversity by +improving the efficiency of conservation efforts that rely +on distributed sensing networks. One such instance where +TinyML has been deployed is in Asia and Africa to resolve +human-elephant conflict. By only transmitting notifications +of elephant detection instead of full video streams to the +cloud, RESOLVE’s WildEyes AI camera can run for more +than 1.5 years on a single Lithium-Ion battery [10]. AI on the +edge is also being used at Liwonde National Park in Malawi +to prevent poaching, and as of September 2019, the park had +lost 0 animals in 30 months [34]. Similar systems are being +used to prevent collisions with whales in busy waterways. +Google deployed a TinyML model on hydrophones (under- +water microphones) using 1,800 hours of underwater audio +recordings to alert ships in the Vancouver Bay [18]. +Due to the low computational requirements, opportuni- +ties also exist for upcycled and recycled electronic devices +for TinyML applications. Rainforest Connection (RFCx) uses +recycled smartphones to develop solar-powered listening de- +vices for pinpointing deforestation over long distances [27]. +Similar opportunities exist for upcycling MCUs. +3.3 +Climate Action (SDG #13) +Take urgent action to combat climate change and its impacts. +TinyML is well-suited to efforts aimed at combating cli- +mate change and its impacts through environmental mon- +itoring applications. For example, Ribbit Network recently +launched an effort to crowdsource the world’s largest green- +house gas emissions dataset through distributed intelligent +sensors. This network has enabled more cheap, accurate, +and actionable local data on emissions than existing large- +scale, expensive satellite solutions. Similarly, the SmartForest +project utilizes a remote monitoring system to provide infor- +mation on tree growth. This system replaced the need for +150 − 160 employees to regularly go into the field with a sin- +gle trip by a small team to install the sensors [9], significantly +reducing human impact on the ecosystem and increasing +data quality for conservation efforts. +In the long term, TinyML also has the potential to power +the next generation of tiny robots to help reduce the global +impact of climate change. For example, climate change has +contributed to the widespread decline of essential pollinators +like bumble bees [29], threatening the global food supply +(SDG #2 mentioned above). TinyML can help provide intelli- +gence to tiny robots like the Robobee [39] that can be used +as artificial pollinators. However, there are still many chal- +lenges and opportunities to unlock tiny robot learning [25]. +Finally, one area of broad interest is the building sector. +Existing systems that control lighting, automated window +Figure 2. Breakdown of global CO2 emissions as of 2019 [14]. +shading, and HVAC based on occupancy and light intensity +sensors show a 20-40% reduction in building energy usage [1, +22]. Adding ML capabilities to these systems would lead to +further improvements in efficiency. This increased efficiency +is critical as energy production along with residential and +commercial energy usage are leading sectors contributing +to global greenhouse gas emissions (see Figure 2). +4 +Quantifying the Sustainability of +TinyML +The benefits of ML on microcontrollers for environmental +sustainability and beyond will continue to fuel the Inter- +net of Things (IoT) revolution, connecting billions of devices +around us. However, embedding smart computing into every- +day objects may have looming environmental consequences +through increased electronic waste [13]. To better under- +stand the environmental costs associated with TinyML, a +life cycle analysis (LCA) of the complete TinyML system +(i.e., MCU plus peripherals and power supply) is performed. +This analysis demonstrates that the footprint of MCUs and +TinyML systems individually is relatively small. When this +analysis is expanded to consider the global scaled impact +of TinyML, the impact could be substantial if not offset by +using TinyML for sustainable applications. +4.1 +Growing Environmental Risks of IoT Trash +Electronic waste (e-waste) is a growing concern and pollut- +ing our environment. In 2019, it was reported that e-waste +had grown by 20% over the past five years [36], and by +2030, forecasts predict a total of 75 million metric tons of e- +waste [32]. In addition to the e-waste, the carbon emissions +from manufacturing and operating these devices are also +growing and impacting the environment. TinyML has the +potential to drive more demand for innovative IoT solutions +that would advance the ubiquitous computing movement, +but further exacerbate the growing “Internet of Trash.” +3 + +1% +2% +6% +6% +42% +24% +19% +Global CO, Emissions by Sector +Electricity and Heat Production +Industry +Transportation +Residential +Agriculture +Commercial and Public Services +OtherFigure 3. Four different environmental indicators measuring the impact of MCUs on our environment. Each footprint contains +both the operational and embodied footprint of the device, including the five-stage life cycle of an MCU. Data courtesy of +STMicroelectronics [33]. The data from other MCU providers follow the same operational and embodied footprint trends. +Parallels can be drawn from the plastic pandemic. An +abundance of resources (e.g., plastic, silicon) has made it +easy to manufacture “infinitely" at scale. The convenience +offered by the respective products made it easy to ignore +or defer the environmental concerns. As such, plastic has +contributed significantly to land and water pollution, and its +production contributes to global warming by emitting green- +house gases. Plastic also contains toxic chemicals that can +leach into food and water and have been linked to various +health problems. These adverse side effects of a transforma- +tive technology provide a cautionary tale and motivation to +carefully consider the total net benefit of TinyML systems +and applications. +4.2 +Environmental Impact of MCUs +The TinyML life cycle analysis starts at the MCU level with +publicly accessible data from STMicroelectronics [33].1 The +hardware life cycle of an MCU can typically be broken down +into five stages: 1) extraction and treatment of raw materi- +als, 2) product manufacturing, 3) transport and distribution, +4) product use, and 5) end of life. Taking these stages into +account, there are four different environmental indicators, +1The general trends hold for other MCU manufacturers when comparing +operational and embodied footprints. +as shown in Figure 3, that can be used to analyze the foot- +print of the processing hardware required for TinyML: water +demand, freshwater eutrophication, photochemical oxidant +formation, and climate change. Across all four indicators, +production, or more specifically, energy consumption during +production, is the dominant driver of an MCU’s environmen- +tal footprint, as noted in previous work [19, 40]. However, +the exact breakdown varies across indicators. +Water Demand. SDG #6 highlighted that billions of peo- +ple are without abundant access to clean water. This indicator +measures the volume of water evaporated, consumed, used +for cooling, or released downstream, during an MCU’s life cy- +cle. Figure 3 shows that while much (54%) of the water used +in an MCU’s life cycle is attributed to the production site, +extracting and transforming the raw materials also requires +a substantial amount of water (41%). +Freshwater Eutrophication. Eutrophication, the prolif- +eration of algae and plants in bodies of water, is one of the +most significant threats to our aquatic ecosystems (SDG #14). +This indicator measures this impact in grams of phosphorous +equivalent (g P-eq.), as phosphorous is a common cause of +algae blooms from over-enriched aquatic ecosystems. Fig- +ure 3 shows that this environmental indicator has the most +balanced impact on the five stages of an MCU life cycle, with +4 + +390g CO2-eq +Total Impact +23L +120mg P-eq +823mg NMVOC +23 bottles +0.2 washing +1.6km +2.7km +0 +by car +of water +cycles +by car +100% +90% +80% +70% +60% +50% +40% +30% +20% +10% +0% +Protochemical +Climate +Water +Freshwater +Oxidant +Change +Demand +Eutrophication +Formation +I End of Life +<1% +<1% +<1% +<1% +? +Logistics +1% +<1% +<1% +1% +IUse +8% +6% +28% +8% +Raw Materials +10% +41% +27% +10% +Production: Other +25% +15% +18% +2% +Production: Energy +56% +39% +27% +71% +Consumption45% of the footprint attributable to production, 28% to the +MCU use, and 27% to the extraction of raw materials. +Photochemical Oxidant Formation. This indicator mea- +sures milligrams of non-methane volatile organic compounds +(NMVOC) formed. These play an essential role in the for- +mation of photochemical oxidants, which can exacerbate +respiratory ailments and lead to smog formation, impacting +the climate (SDG #13), local air quality (SDG #15), and public +health. Figure 3 shows that this footprint is mainly driven by +production, accounting for 74% of the total, with 71% coming +from energy usage during production. +Climate Change. This indicator measures equivalent grams +of carbon dioxide (CO2-eq) emitted. CO2 is the most preva- +lent greenhouse gas produced by humans and a primary +driver of climate change (SDG #13). As Figure 3 shows, most +of the carbon emissions come during production of the MCU +(81%), with the majority resulting from energy consumption +(56%). The entire carbon footprint of an MCU is 390 g CO2-eq. +For perspective, this footprint is equivalent to a gasoline- +powered car driving 1.6 km. Given that cars typically drive +hundreds of thousands of miles over their lifetime, a single +MCU alone has minimal impact in the context of everyday +human actions. In the following section, CO2 emissions are +used as the primary measure due to their wide acceptance +for assessing environmental impact. +4.3 +Footprint of TinyML Systems +MCUs are the heart of embedded TinyML systems, but we +must consider the additional components that constitute a +complete TinyML system to get a more accurate picture of +the complete footprint. Thus, in this section, we systemati- +cally analyze the footprint of the systems used for the widely +deployed TinyML applications of keyword spotting and im- +age classification. This analysis outlines all pieces needed +for deploying a system in the wild such as casing, sensing, +actuators, transport, and more. +TinyML Footprint Calculator. We developed an open- +source TinyML Footprint Calculator to evaluate the foot- +print of TinyML systems.2 This tool can be used in the fu- +ture to help engineers understand the impact of the devices +they are developing. For example, this tool could be used +to produce the environmental impact report for ML sensor +datasheets [37]. +Our calculator leverages the raw data from a recent 2021 +study by Pirson and Bol [26] assessing the embodied carbon +footprint of IoT devices. Pirson and Bol [26] break down +the general architecture and hardware profile of an IoT edge +device into a collection of basic functional blocks: processing, +memory, actuators, casing, connectivity, PCB, power supply, +2Code, documentation, and a link to the online calculator can be found at +https://github.com/harvard-edge/TinyML-Footprint +security, sensing, transport, user interface, and others circuit +components (e.g., resistors, capacitors, diodes). +Within the various blocks, Pirson and Bol break down +the impact based on application specifications. However, +Pirson and Bol note that the data provided only encapsulates +stages 1 through 3 of the hardware life cycle (i.e., embodied +footprint). As such, we additionally model and capture the +product use stage (i.e., operational footprint) and end-of-life +stage of the hardware life cycle in Figure 4. +To account for the use stage, we calculated the CO2-eq +of recharging a power supply for three years of continuous +use at 1 mW. This number is an average estimate of the +power used by commercial TinyML systems as reported in +the MLPerf Tiny Benchmark [3] energy results [23]. We note +that some TinyML applications may require much less power +than 1 mW when idle (e.g., keyword spotting) and that we +used a three-year benchmark to be consistent with Apple’s +analysis of their hardware which was used as a baseline. +In addition, we included the ML model training costs since +they can be large, on the order of millions of kg of CO2-eq +for large cloud ML tasks [40]. Costs were based on footprint +estimates of DenseNet [11], which serve as an upper bound +on computation because it exceeds the typical size of TinyML +models. +Breaking Down TinyML’s Footprint. The calculated foot- +print of TinyML systems is broken down into three scenarios. +The “Low-Cost Profile” scenario represents a keyword spot- +ting application that requires only a simple microphone sen- +sor. The “Medium-Cost Profile” scenario represents an image +classification application that requires a much larger camera +sensor. The “High-Cost Profile” scenario again uses the image +classification application, instead using the upper bound car- +bon emission values for each component provided in Pirson +and Bol [26]. These three scenarios represent typical upper +and lower bound footprints for assessing classical TinyML +systems but are not absolute bounds. For complete details +of our setup, see https://github.com/harvard-edge/TinyML- +Footprint. +As the stacked bar graph on the right side of Figure 4 +shows, the embodied footprint of all components is much +greater than the system’s operational footprint (captured +in “Product Use"). This result aligns with previous literature +suggesting that manufacturing dominates the environmental +footprint of small electronics [19, 40]. Moreover, the figure +highlights that the processing’s (i.e., MCU) embodied foot- +print does not contribute significantly to the overall footprint. +Instead, most of the footprint is attributable to the embodied +footprint of the additional components (e.g., power supply, +sensor, circuit board) and the transportation costs associated +with manufacturing and distribution. In particular, the power +supply is one of the dominating factors in the footprint. The +embodied footprint of a battery required to deploy TinyML +5 + +Figure 4. A breakdown of different TinyML system footprints highlights that the footprint is largely attributable to the +embodied footprint of the power supply, onboard sensors, and transportation. Note that actuator and connectivity blocks from +Pirson and Bol [26] are encapsulated in “Other" and “Processing", respectively, while “Product Use" captures the operational +footprint. The carbon footprint of Apple’s Series 7 Watch [16] and 16-inch MacBook Pro [15] are also provided for reference. For +more details and to compute the footprint of your own TinyML system, see https://github.com/harvard-edge/TinyML-Footprint. +in the wild for years is much larger than any other system +component. +One may consider comparing our TinyML system foot- +print to another device used for making progress toward +the SDGs or an edge-class server. However, the documen- +tation of such data is still relatively new and limited. Thus, +using what is publicly available, the typical TinyML system +footprint is compared with the latest Apple Watch Series 7 +(representative of an “edge" device) to provide a baseline +reference for understanding the total carbon footprint of a +TinyML device, as shown on the left of Figure 4. In the figure, +the footprint of a 16-inch MacBook Pro is also provided to +give the reader an idea of a device footprint representative +of traditional computing hardware. +The carbon footprint of the Apple Watch, considering +three years of use, is 34 kg CO2, with 76% of the footprint +attributable to production, 10% to transport, and 13% to every- +day use [16]. This carbon footprint is 5-38× larger compared +to a TinyML system. Moreover, for reference, a TinyML sys- +tem has a 49-392× smaller footprint than a Macbook [15]. +This complete LCA shows that the additional components +that constitute a TinyML system have a larger carbon foot- +print than the MCU alone. However, TinyML systems still +have a much smaller footprint than existing edge or tradi- +tional computing devices. +4.4 +TinyML at Scale +Each TinyML device will have an associated environmental +footprint, as outlined in Sections 4.2 and 4.3, and can provide +environmental benefits, as discussed in Section 3. However, +humanity is on the path to a future with billions of deployed +intelligent IoT devices. To better understand the net effect +of TinyML at scale, this section assesses what happens to +TinyML’s footprint if these systems are scaled to the number +of MCUs deployed globally, which currently sits at around +250 billion. While including all 250 billion existing MCUs in +TinyML’s global footprint may not be necessary, as likely +only a fraction of them will run TinyML, this aims to provide +an upper bound for the net effect of TinyML both now and +into the near future. To continue this upper bound assump- +tion, the “High-cost Profile” data from Figure 4 is used. This +scenario results in a combined, non-trivial global carbon +footprint of 1765 million metric tons of CO2-eq. +1765 million metric tons of CO2-eq is a quite concern- +ing footprint for TinyML on its own. However, along with +this number we need to consider the emissions that were +avoided by using these systems to fairly evaluate the com- +plete environmental impact of TinyML. As mentioned earlier, +there are existing examples (e.g., [1, 22]) of simple, intelligent +IoT devices which can reduce building CO2 emissions by at +least 20%. In this case, enduring the footprint of these smart +devices is worthwhile as their footprint is most likely neg- +ligible when compared to the 20% reduction in a building’s +emissions they provided. +Figure 5 now compares the calculated global carbon foot- +print of TinyML (blue bar) in the context of the emissions +these TinyML systems could help avoid through efficiency +improvements in other sectors. If the aforementioned 20% +6 + +10x +Other (e.g., Product Use, End of Life) +7.00 +ML Training +100 + Processing (e.g., MCU, Memory) +6.00 +Casing, PCB, and User Interface + Transportation +5x +10x +38x +5.00 + Sensing Module +10 + Power Supply (3 years) +4.00 +3.00 +1 +2.00 +1.00 +0.1 +16-inch +Apple +TinyML +TinyML +TinyML +Medium- +0.00 +MacBook +Watch +High- +Low- +TinyML +TinyML +TinyML +Pro +Series 7 +Cost +Cost +Cost +High-Cost +Medium-Cost +Low-Cost +349 +34 +7.1 +3.4 +0.9Figure 5. If all 250 billion MCUs were TinyML systems with three-year lifespans, their worst-case footprint would be 1765 +million metric tons of CO2. Fortunately, if these systems enabled a 20% reduction in emissions for the residential sector and +only a 0.6% reduction in all other sectors (Figure 2), the total footprint would be net-zero. Anything larger, 20% shown as an +example, results in more carbon savings from TinyML than emissions. +reduction in a single building’s emissions were applied to +the entire residential home sector over three years (green +bar), 1181 million metric tons of CO2-eq would be avoided. +These avoided emissions alone would offset 67% of the worst- +case costs of TinyML. The residential sector, though, only +represents 6% of total global emissions (Figure 2). If remain- +ing TinyML devices were able to reduce emissions from all +other sectors by as little as 0.6% on average (orange bar), then +TinyML would break even from an emissions standpoint. +Furthermore, if we were to extrapolate this 20% reduction +in the residential sector to all sectors (yellow bar) we would +see a net reduction in global CO2 emissions by over 18.4 +billion metric tons. While a 20% reduction in emissions from +all sectors may be unrealistic, anything greater than a 0.6% +reduction on average would result in TinyML saving more +emissions than it produced. This result suggests that TinyML +systems, if designed with careful intention, can elicit an +overall positive impact on the environment. +4.5 +Limitations of Our Study +In this section, we recognize the limitations of our analysis. +While our research provides a starting point, more needs +to be done to fully understand the impact of TinyML on +environmental sustainability. +One major limitation is the lack of publicly available data +on the environmental impact of modern digital electronics. +This makes it difficult for our analysis to be detailed and pre- +cise, and also makes it challenging for consumers to make +informed decisions about the environmental impact of their +purchases. However, it is promising to see that there is in- +creasing demand for LCA and carbon footprint data in the +information and communications technology industry. This +additional data will empower consumers and make it feasible +to comprehensively account for the heterogeneity in TinyML +and IoT systems in future work. +Another important consideration is Jevons paradox, which +suggests that advancements in efficiency can lead to an over- +all increase in consumption and a negative impact on the +environment. Section 4.4 attempts to address this by exam- +ining a scenario in which TinyML systems are produced at +a large scale (i.e., 250 billion TinyML systems) to improve +the efficiency and sustainability of other sectors. This anal- +ysis suggests that TinyML systems have the potential to +have a positive impact on the environment. However, the +conclusions of this assessment are limited by the current +data limitations, and more research is needed to make more +accurate predictions. +5 +Future Sustainable TinyML +While TinyML has the potential to contribute to global sus- +tainable development and environmental sustainability, there +are still many challenges that must be overcome to fully real- +ize this potential. As we have shown in Section 4.4, the envi- +ronmental impact of TinyML will be non-negligible. Even if +the benefits of TinyML can potentially outweigh its impact, +it is important to be cautious and ensure that future genera- +tions of TinyML devices are sustainable. In this section, we +will discuss the broader implications of our study and suggest +ways to make TinyML more sustainable. It should be noted +that for sustainable design to be truly successful, though, +incentives must be provided to corporations and engineers +to prioritize sustainability when making decisions. +Energy Harvesting. Our analysis in Section 4.3 revealed +that the batteries used to power TinyML devices dominate +their environmental impact. Moreover, batteries present sev- +eral other environmental issues, such as water and air pollu- +tion and the release of carcinogens [20]. Research in energy +harvesting [28] could reduce the batteries needed and the +associated environmental costs that come with it. Further- +more, advancements in intermittent computing [12] could +7 + +2000 +1765 +(1181) +1000 +(584) +(18,991) +0 +Total TinyML +20% Savings in +0.6% Savings in all +20% Savings in all +Footprint +Residential Sector +Other Sectors +Other Sectors +-1000 +High-Cost +(3 Year) +(3 Years) +(3 Years) +-2000 +-18000 +-19000be suitable in TinyML scenarios, which would further reduce +the needed power supply. +Efficient Sensing. The second largest contributor to the +footprint in two of the three profiles analyzed was the sensor. +In these two cases, a camera sensor was used, whereas in +the other case, a microphone was used. Sensing is essential +in TinyML, but using smaller (e.g., camera vs. inertial mea- +surement unit) or lower-quality sensors (e.g., low- vs. high- +resolution camera) could further reduce the environmental +impact of the device. Due to the relatively low footprint +of the compute, more advanced TinyML models could be +used to make up for the loss in performance introduced by +lower-quality sensors, thus potentially reducing the overall +footprint while achieving the same performance. Addition- +ally, sensor fusion, which uses multiple small sensors to +make up for the functionality of one large sensor, could also +be used to reduce the environmental impact of sensing [7]. +Datasheets for ML Sensors. Greater transparency re- +garding the system’s data and costs is needed to deploy +these TinyML devices safely and ethically. TinyML instantia- +tions must clearly and transparently articulate their privacy +and security boundaries. One solution to address privacy +concerns is to separate the input sensor data and ML pro- +cessing from the rest of the system at the hardware level. +Also, new supplementary information is needed in the form +of a datasheet that builds upon traditional datasheets used +for electrical components to enable transparency to end +users [37]. These datasheets should include information +about the environmental impact and LCA of the device in +an easy-to-understand format so that users can use TinyML +devices in an environmentally friendly and sustainable way. +Datasets for Low-Resource Domains. Many TinyML +applications involve embedded sensors and require real- +world data, which can be difficult to find, especially in public +domains. There is a need for large, open-access datasets that +focus on low-resource, high-impact problems involving sen- +sors. Thus, to support TinyML development, it is necessary +to create a dataset similar to ImageNet for TinyML. Just as +open-source software development has allowed the industry +to share code and reduce development costs, a shift is needed +in thinking about sharing real data to create large, public, +and representative datasets that can support TinyML use +cases and applications. +Furthermore, while collecting data for TinyML is vitally +important, it is also important to consider the environmental +impact of the data collection. For example, collecting data in +nature can be disruptive and harmful to habitats as data may +only be available in remote locations that require travel (e.g., +boats or aircraft). On the other hand, alliterative methods +such as computationally-intensive simulations may require +carbon-consuming resources to obtain. Some of these costs +may have to be incurred to provide a larger benefit to these +environments in the long term, but these costs should be +minimized. Data sharing needs to be encouraged so that +collection has to happen as few times as possible. +Emerging Technologies. New technologies are being de- +veloped that could lead to more sustainable TinyML practices. +One example is flexible electronics, which has demonstrated +a 1000× smaller carbon footprint compared to traditional +silicon manufacturing [5]. Such technologies could enable +TinyML to achieve greater reductions in emissions than an- +ticipated in our analysis. However, these technologies are +not yet mature and have less processing power than tradi- +tional silicon devices [4]. More research and development is +needed to utilize these sustainable technologies fully. +Recycle and Upcycle. TinyML can potentially exacer- +bate the problem of electronic waste. However, recycling +and reusing TinyML devices is a viable option as many of +the algorithms can run on standard, commonly used MCU +hardware. This can extend the life of the MCU and reduce +the amount of waste sent to landfills. It’s worth noting that +in our analysis of TinyML systems in Section 4.3, a three- +year lifetime was assumed as a fair comparison with other +LCA data from Apple. However, in reality, TinyML systems +can often last closer to 10 years, which would further reduce +their environmental impact over time. +Accessibility. Finally, for TinyML to have a significant +impact on a global scale, it takes more than just technology. +There is a need for global access to hardware and educational +resources. Fortunately, there have been some recent efforts, +led by the TinyML foundation3 and the TinyML Open Educa- +tion Initiative (TinyMLedu)4, among others, to both develop +such open-source materials and provide low-cost or no-cost +hardware to learners [17]. +6 +Conclusion +Using ML on microcontrollers can have a significant impact +on environmental sustainability. Low-power ML on low-cost +MCU-class hardware has the potential to improve efficiency +in various sectors, enabling significant reductions in carbon +emissions. This assessment shows that TinyML’s carbon +footprint could be offset by using the technology to reduce +emissions from other economic sectors. However, TinyML’s +footprint is not negligible when scaled globally, and thus +designers must be mindful and factor in sustainability when +developing new devices. 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Proceedings of Machine Learning and +Systems 4 (2022), 795–813. +10 + diff --git a/jdFKT4oBgHgl3EQfwi40/content/tmp_files/load_file.txt b/jdFKT4oBgHgl3EQfwi40/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..39aca840d1cd981459dfe7eafc3a809be63bd0a8 --- /dev/null +++ b/jdFKT4oBgHgl3EQfwi40/content/tmp_files/load_file.txt @@ -0,0 +1,753 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf,len=752 +page_content='Is TinyML Sustainable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Assessing the Environmental Impacts of Machine Learning on Microcontrollers Shvetank Prakash Harvard University sprakash@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='edu, USA Matthew Stewart Harvard University matthew_stewart@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='edu USA Colby Banbury Harvard University cbanbury@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='edu, USA Mark Mazumder Harvard University markmazumder@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='edu, USA Pete Warden Stanford University petewarden@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='edu, USA Brian Plancher Barnard College, Columbia University bplancher@barnard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='edu, USA Vijay Janapa Reddi Harvard University vj@eecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='edu, USA Abstract The sustained growth of carbon emissions and global waste elicits significant sustainability concerns for our environ- ment’s future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The growing Internet of Things (IoT) has the potential to exacerbate this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, an emerging area known as Tiny Machine Learning (TinyML) has the opportu- nity to help address these environmental challenges through sustainable computing practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' TinyML, the deployment of machine learning (ML) algorithms onto low-cost, low-power microcontroller systems, enables on-device sensor analyt- ics that unlocks numerous always-on ML applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This article discusses the potential of these TinyML applications to address critical sustainability challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Moreover, the footprint of this emerging technology is assessed through a complete life cycle analysis of TinyML systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' From this analysis, TinyML presents opportunities to offset its carbon emissions by enabling applications that reduce the emis- sions of other sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Nevertheless, when globally scaled, the carbon footprint of TinyML systems is not negligible, necessitating that designers factor in environmental impact when formulating new devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Finally, research directions for enabling further opportunities for TinyML to contribute to a sustainable future are outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 1 Introduction The continued growth of carbon emissions and global waste presents a great concern for our environment, increasing calls for a more sustainable future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' In response, the United Nations’ (UN) 2030 Agenda for Sustainable Development established a shared framework aiming toward peace and prosperity for people and the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' At its core are 17 Sus- tainable Development Goals (SDGs) [36], a call to action for all countries to work towards a more environmentally, economically, and socially sustainable future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Tiny machine learning (TinyML), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' the ability to run ML on microcontroller devices, unlocks an expansive new array of always-on ML applications that can help address many of Maintain Design Deploy TinyML Cycle Manufacturing Energy Raw Materials Microcontrollers Logistics Costs End-of-life Recycling Upcycling E-waste Energy Consumption Sustainable Development Goals Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' TinyML can contribute to the UN’s environmen- tal sustainability goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, the life cycle of TinyML systems must be considered to maximize their global ben- efit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The positive (green arrows) and negative (red arrows) environmental footprint of TinyML systems are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' the UN’s SDGs, especially those that target environmental sustainability (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Often, only the operational ben- efits of TinyML are noted for environmental sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, in this article we argue that the entire life cycle of both TinyML applications and hardware must be considered to ensure that these new technologies provide more carbon savings than emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' To this end, the contributions of the paper are the following: (1) an assortment of case studies demonstrating TinyML’s sustainability benefits is provided, (2) the environmental impacts of TinyML are outlined at the individual MCU-level and at the complete, integrated TinyML system-level through a life cycle analysis, and (3) future work is identified in enabling sustainable TinyML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Section 2 provides background on TinyML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Section 3 explores sus- tainable applications enabled by TinyML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Section 4 uses a complete life cycle analysis to assess the environmental foot- print of a single MCU and complete TinyML system along arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='11899v1 [cs.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='SUFFICIENT OXYGEN TO SUPPORT MARINELIFE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='FISHERY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='OCEAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='OVERHALFOF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='COLLAPSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='WARMING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='MARINEKEY BIODIVERSITYAREAS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='ARENOTPROTECTED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='CO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='ACIDIFICATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='EUTROPHICATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='OVER3BILLIONPEOPLE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='RELY ON OCEANSFORTHEIRLIVELIHOODS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='ABOUTHALFOF COUNTRIESWORLDWIDE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='ON AVERAGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='0NLY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='2% HAVEADOPTED SPECIFICINITIATIVES OFNATIONALRESEARCHBUDGETSARE TO SUPPORT SMALL-SCALEFISHERS ALLOCATED FOR OCEAN SCIENCE 333333 5333333 THESUSTAINABLEDEVELOPMENTG0ALSREPORT2021:UNSTATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='UN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='0RG/SDGS/REPORT/2021/with the impact at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Finally, future research directions for Sustainable TinyML are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 2 Tiny Machine Learning (TinyML) TinyML is the deployment of machine learning (ML) algo- rithms onto low-cost, low-power, and resource-constrained MCU systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' TinyML stores neural network models directly within memory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', flash) and runs inference directly on the output of onboard sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This approach enables intelligent on-device sensor analytics unavailable with traditional Inter- net of Things (IoT) approaches, which instead typically rely on communication with the cloud to transmit data for exter- nal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Importantly, TinyML achieves this using a fraction of the compute resources needed for traditional ML systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Table 1 compares TinyML with traditional BigML (such as cloud and mobile systems) and shows how TinyML requires orders of magnitude fewer resources across com- pute, memory, storage, power, and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Finally, while the heterogeneity and limited resources of MCU devices present new challenges for on-device training, model updating, and deployment, recent research and the development of ML frameworks such as TensorFlow Lite for Microcontrollers [8] have increased the accessibility of TinyML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The ubiquity, low-cost, and small power envelope of MCUs, paired with TinyML’s independence from internet connec- tivity, enables ML models to be deployed globally anywhere at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' For these reasons, along with bandwidth, latency, energy, reliability, and privacy concerns, running ML directly on these embedded edge devices is growing in popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' With more than 250 billion MCUs deployed globally today, and the cost of MCUs expected to drop below $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='50 per unit, this number is expected to grow, eclipsing 50 billion MCUs shipped per annum in the next decade [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' As such, TinyML will become an ever-present technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' But the question we must ask ourselves is do we run the risk of producing an Internet of Trash over the course of TinyML devices’ lifetime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 3 Applications of TinyML for Sustainability To fairly evaluate the environmental impacts of machine learning on microcontrollers, we first consider TinyML’s ben- efits, given that the creation of value is what will likely result in widespread TinyML deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Typical well-known consumer-facing applications of TinyML include keyword spotting, image classification, and anomaly detection [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, many other applications of TinyML can be used to enable a more sustainable future [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' In the following sec- tions, emerging applications are highlighted which show how TinyML can help make progress towards important environmental-related SDGs (as shown in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' In par- ticular, TinyML is shown to be well-suited for improving the sustainability of global agriculture, aiding wildlife conserva- tion, and helping combat climate change and its impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Platform Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Memory Storage Power Price CO2-eq Footprint Cloud GHz 10+GB TBs-PBs ∼1 kW $1000+ Hundreds of kgs Mobile GHz Few GB GBs ∼1 W $100+ Tens of kgs Tiny MHz KBs Few MB ∼1 mW $10 Single kgs Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Cloud and mobile ML systems are compared to TinyML systems (in bold) across frequency, memory, store, power, price, and footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This work shows the footprint of TinyML systems is on the order of a few kilograms of CO2-eq, far less than cloud and mobile ML systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='1 Zero Hunger & Good Health and Well-Being (SDG #2 & #3) End hunger, achieve food security & improved nutrition, pro- mote sustainable agriculture, and ensure healthy lives while promoting well-being for all at all ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' ML applications can increase agriculture production through data-driven methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' For example, Nuru, a mobile and cloud- based ML app from the PlantVillage project, is more accu- rate than humans at detecting plant diseases and enabled one farmer to increase her revenue by 55% and yields by 146% [6, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' ML has also been used for autonomous devices such as tiny drones, which can provide targeted pesticide applications that reduce use to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='1% of conventional blanket spraying [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' As another example, researchers developed a cough monitor system to flag respiratory problems in pigs by placing microphones over animal pens which can alert farmers 12 days earlier than standard methods [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' TinyML has the potential to increase the impact of these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' First, it can enable these and many other appli- cations to be used in remote regions through low-power, low-connectivity operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Second, it can enable scaled deployment of these smart sensors, which could provide more targeted information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', on all individual pigs in real- time for the cough monitor system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Most importantly, it would increase global access to these technologies by reduc- ing costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' As Sparrow and Howard note, global adoption can only occur if devices “can be manufactured and sold cheaply enough to be available to smaller farms” [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' On another very important and serious note, TinyML can also be used to aid in our health and well-being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' One of the diseases noted in the UN’s SDG report when considering our well-being is malaria due to its massive global impact that spans a long history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' In fact, nearly half of the whole world population has been killed by mosquitoes [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Gaps in fund- ing and access to life-saving tools led to a disproportionate 94% of all malaria cases and deaths in 2019 occurring in the African region [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Using Edge Impulse, a development plat- form for TinyML, a system was prototyped to identify the deadliest mosquitoes using wing beats sound classification with 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='3% accuracy [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This is another example in which global access to these systems will have a tremendous impact and could potentially save lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='2 Life on Land & Below Water (SDGs #14 & #15) Protect, restore, conserve, and promote sustainable use of ter- restrial & aquatic ecosystems, sustainably manage forests & marine resources, combat desertification, halt biodiversity loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' TinyML can help preserve the planet’s biodiversity by improving the efficiency of conservation efforts that rely on distributed sensing networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' One such instance where TinyML has been deployed is in Asia and Africa to resolve human-elephant conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' By only transmitting notifications of elephant detection instead of full video streams to the cloud, RESOLVE’s WildEyes AI camera can run for more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='5 years on a single Lithium-Ion battery [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' AI on the edge is also being used at Liwonde National Park in Malawi to prevent poaching, and as of September 2019, the park had lost 0 animals in 30 months [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Similar systems are being used to prevent collisions with whales in busy waterways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Google deployed a TinyML model on hydrophones (under- water microphones) using 1,800 hours of underwater audio recordings to alert ships in the Vancouver Bay [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Due to the low computational requirements, opportuni- ties also exist for upcycled and recycled electronic devices for TinyML applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Rainforest Connection (RFCx) uses recycled smartphones to develop solar-powered listening de- vices for pinpointing deforestation over long distances [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Similar opportunities exist for upcycling MCUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='3 Climate Action (SDG #13) Take urgent action to combat climate change and its impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' TinyML is well-suited to efforts aimed at combating cli- mate change and its impacts through environmental mon- itoring applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' For example, Ribbit Network recently launched an effort to crowdsource the world’s largest green- house gas emissions dataset through distributed intelligent sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This network has enabled more cheap, accurate, and actionable local data on emissions than existing large- scale, expensive satellite solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Similarly, the SmartForest project utilizes a remote monitoring system to provide infor- mation on tree growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This system replaced the need for 150 − 160 employees to regularly go into the field with a sin- gle trip by a small team to install the sensors [9], significantly reducing human impact on the ecosystem and increasing data quality for conservation efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' In the long term, TinyML also has the potential to power the next generation of tiny robots to help reduce the global impact of climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' For example, climate change has contributed to the widespread decline of essential pollinators like bumble bees [29], threatening the global food supply (SDG #2 mentioned above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' TinyML can help provide intelli- gence to tiny robots like the Robobee [39] that can be used as artificial pollinators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, there are still many chal- lenges and opportunities to unlock tiny robot learning [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Finally, one area of broad interest is the building sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Existing systems that control lighting, automated window Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Breakdown of global CO2 emissions as of 2019 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' shading, and HVAC based on occupancy and light intensity sensors show a 20-40% reduction in building energy usage [1, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Adding ML capabilities to these systems would lead to further improvements in efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This increased efficiency is critical as energy production along with residential and commercial energy usage are leading sectors contributing to global greenhouse gas emissions (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 4 Quantifying the Sustainability of TinyML The benefits of ML on microcontrollers for environmental sustainability and beyond will continue to fuel the Inter- net of Things (IoT) revolution, connecting billions of devices around us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, embedding smart computing into every- day objects may have looming environmental consequences through increased electronic waste [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' To better under- stand the environmental costs associated with TinyML, a life cycle analysis (LCA) of the complete TinyML system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', MCU plus peripherals and power supply) is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This analysis demonstrates that the footprint of MCUs and TinyML systems individually is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' When this analysis is expanded to consider the global scaled impact of TinyML, the impact could be substantial if not offset by using TinyML for sustainable applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='1 Growing Environmental Risks of IoT Trash Electronic waste (e-waste) is a growing concern and pollut- ing our environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' In 2019, it was reported that e-waste had grown by 20% over the past five years [36], and by 2030, forecasts predict a total of 75 million metric tons of e- waste [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' In addition to the e-waste, the carbon emissions from manufacturing and operating these devices are also growing and impacting the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' TinyML has the potential to drive more demand for innovative IoT solutions that would advance the ubiquitous computing movement, but further exacerbate the growing “Internet of Trash.” 3 1% 2% 6% 6% 42% 24% 19% Global CO, Emissions by Sector Electricity and Heat Production Industry Transportation Residential Agriculture Commercial and Public Services OtherFigure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Four different environmental indicators measuring the impact of MCUs on our environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Each footprint contains both the operational and embodied footprint of the device, including the five-stage life cycle of an MCU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Data courtesy of STMicroelectronics [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The data from other MCU providers follow the same operational and embodied footprint trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Parallels can be drawn from the plastic pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' An abundance of resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', plastic, silicon) has made it easy to manufacture “infinitely" at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The convenience offered by the respective products made it easy to ignore or defer the environmental concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' As such, plastic has contributed significantly to land and water pollution, and its production contributes to global warming by emitting green- house gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Plastic also contains toxic chemicals that can leach into food and water and have been linked to various health problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' These adverse side effects of a transforma- tive technology provide a cautionary tale and motivation to carefully consider the total net benefit of TinyML systems and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='2 Environmental Impact of MCUs The TinyML life cycle analysis starts at the MCU level with publicly accessible data from STMicroelectronics [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='1 The hardware life cycle of an MCU can typically be broken down into five stages: 1) extraction and treatment of raw materi- als, 2) product manufacturing, 3) transport and distribution, 4) product use, and 5) end of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Taking these stages into account, there are four different environmental indicators, 1The general trends hold for other MCU manufacturers when comparing operational and embodied footprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' as shown in Figure 3, that can be used to analyze the foot- print of the processing hardware required for TinyML: water demand, freshwater eutrophication, photochemical oxidant formation, and climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Across all four indicators, production, or more specifically, energy consumption during production, is the dominant driver of an MCU’s environmen- tal footprint, as noted in previous work [19, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, the exact breakdown varies across indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Water Demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' SDG #6 highlighted that billions of peo- ple are without abundant access to clean water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This indicator measures the volume of water evaporated, consumed, used for cooling, or released downstream, during an MCU’s life cy- cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Figure 3 shows that while much (54%) of the water used in an MCU’s life cycle is attributed to the production site, extracting and transforming the raw materials also requires a substantial amount of water (41%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Freshwater Eutrophication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Eutrophication, the prolif- eration of algae and plants in bodies of water, is one of the most significant threats to our aquatic ecosystems (SDG #14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This indicator measures this impact in grams of phosphorous equivalent (g P-eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' ), as phosphorous is a common cause of algae blooms from over-enriched aquatic ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Fig- ure 3 shows that this environmental indicator has the most balanced impact on the five stages of an MCU life cycle, with 4 390g CO2-eq Total Impact 23L 120mg P-eq 823mg NMVOC 23 bottles 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='2 washing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='6km 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='7km 0 by car of water cycles by car 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Protochemical Climate Water Freshwater Oxidant Change Demand Eutrophication Formation I End of Life <1% <1% <1% <1% ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Logistics 1% <1% <1% 1% IUse 8% 6% 28% 8% Raw Materials 10% 41% 27% 10% Production: Other 25% 15% 18% 2% Production: Energy 56% 39% 27% 71% Consumption45% of the footprint attributable to production, 28% to the MCU use, and 27% to the extraction of raw materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Photochemical Oxidant Formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This indicator mea- sures milligrams of non-methane volatile organic compounds (NMVOC) formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' These play an essential role in the for- mation of photochemical oxidants, which can exacerbate respiratory ailments and lead to smog formation, impacting the climate (SDG #13), local air quality (SDG #15), and public health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Figure 3 shows that this footprint is mainly driven by production, accounting for 74% of the total, with 71% coming from energy usage during production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Climate Change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This indicator measures equivalent grams of carbon dioxide (CO2-eq) emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' CO2 is the most preva- lent greenhouse gas produced by humans and a primary driver of climate change (SDG #13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' As Figure 3 shows, most of the carbon emissions come during production of the MCU (81%), with the majority resulting from energy consumption (56%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The entire carbon footprint of an MCU is 390 g CO2-eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' For perspective, this footprint is equivalent to a gasoline- powered car driving 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='6 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Given that cars typically drive hundreds of thousands of miles over their lifetime, a single MCU alone has minimal impact in the context of everyday human actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' In the following section, CO2 emissions are used as the primary measure due to their wide acceptance for assessing environmental impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='3 Footprint of TinyML Systems MCUs are the heart of embedded TinyML systems, but we must consider the additional components that constitute a complete TinyML system to get a more accurate picture of the complete footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Thus, in this section, we systemati- cally analyze the footprint of the systems used for the widely deployed TinyML applications of keyword spotting and im- age classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This analysis outlines all pieces needed for deploying a system in the wild such as casing, sensing, actuators, transport, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' TinyML Footprint Calculator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' We developed an open- source TinyML Footprint Calculator to evaluate the foot- print of TinyML systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='2 This tool can be used in the fu- ture to help engineers understand the impact of the devices they are developing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' For example, this tool could be used to produce the environmental impact report for ML sensor datasheets [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Our calculator leverages the raw data from a recent 2021 study by Pirson and Bol [26] assessing the embodied carbon footprint of IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Pirson and Bol [26] break down the general architecture and hardware profile of an IoT edge device into a collection of basic functional blocks: processing, memory, actuators, casing, connectivity, PCB, power supply, 2Code, documentation, and a link to the online calculator can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='com/harvard-edge/TinyML-Footprint security, sensing, transport, user interface, and others circuit components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', resistors, capacitors, diodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Within the various blocks, Pirson and Bol break down the impact based on application specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, Pirson and Bol note that the data provided only encapsulates stages 1 through 3 of the hardware life cycle (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', embodied footprint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' As such, we additionally model and capture the product use stage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', operational footprint) and end-of-life stage of the hardware life cycle in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' To account for the use stage, we calculated the CO2-eq of recharging a power supply for three years of continuous use at 1 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This number is an average estimate of the power used by commercial TinyML systems as reported in the MLPerf Tiny Benchmark [3] energy results [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' We note that some TinyML applications may require much less power than 1 mW when idle (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', keyword spotting) and that we used a three-year benchmark to be consistent with Apple’s analysis of their hardware which was used as a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' In addition, we included the ML model training costs since they can be large, on the order of millions of kg of CO2-eq for large cloud ML tasks [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Costs were based on footprint estimates of DenseNet [11], which serve as an upper bound on computation because it exceeds the typical size of TinyML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Breaking Down TinyML’s Footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The calculated foot- print of TinyML systems is broken down into three scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The “Low-Cost Profile” scenario represents a keyword spot- ting application that requires only a simple microphone sen- sor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The “Medium-Cost Profile” scenario represents an image classification application that requires a much larger camera sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The “High-Cost Profile” scenario again uses the image classification application, instead using the upper bound car- bon emission values for each component provided in Pirson and Bol [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' These three scenarios represent typical upper and lower bound footprints for assessing classical TinyML systems but are not absolute bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' For complete details of our setup, see https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='com/harvard-edge/TinyML- Footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' As the stacked bar graph on the right side of Figure 4 shows, the embodied footprint of all components is much greater than the system’s operational footprint (captured in “Product Use").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This result aligns with previous literature suggesting that manufacturing dominates the environmental footprint of small electronics [19, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Moreover, the figure highlights that the processing’s (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', MCU) embodied foot- print does not contribute significantly to the overall footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Instead, most of the footprint is attributable to the embodied footprint of the additional components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', power supply, sensor, circuit board) and the transportation costs associated with manufacturing and distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' In particular, the power supply is one of the dominating factors in the footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The embodied footprint of a battery required to deploy TinyML 5 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' A breakdown of different TinyML system footprints highlights that the footprint is largely attributable to the embodied footprint of the power supply, onboard sensors, and transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Note that actuator and connectivity blocks from Pirson and Bol [26] are encapsulated in “Other" and “Processing", respectively, while “Product Use" captures the operational footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The carbon footprint of Apple’s Series 7 Watch [16] and 16-inch MacBook Pro [15] are also provided for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' For more details and to compute the footprint of your own TinyML system, see https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='com/harvard-edge/TinyML-Footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' in the wild for years is much larger than any other system component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' One may consider comparing our TinyML system foot- print to another device used for making progress toward the SDGs or an edge-class server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, the documen- tation of such data is still relatively new and limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Thus, using what is publicly available, the typical TinyML system footprint is compared with the latest Apple Watch Series 7 (representative of an “edge" device) to provide a baseline reference for understanding the total carbon footprint of a TinyML device, as shown on the left of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' In the figure, the footprint of a 16-inch MacBook Pro is also provided to give the reader an idea of a device footprint representative of traditional computing hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The carbon footprint of the Apple Watch, considering three years of use, is 34 kg CO2, with 76% of the footprint attributable to production, 10% to transport, and 13% to every- day use [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This carbon footprint is 5-38× larger compared to a TinyML system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Moreover, for reference, a TinyML sys- tem has a 49-392× smaller footprint than a Macbook [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This complete LCA shows that the additional components that constitute a TinyML system have a larger carbon foot- print than the MCU alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, TinyML systems still have a much smaller footprint than existing edge or tradi- tional computing devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='4 TinyML at Scale Each TinyML device will have an associated environmental footprint, as outlined in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='3, and can provide environmental benefits, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, humanity is on the path to a future with billions of deployed intelligent IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' To better understand the net effect of TinyML at scale, this section assesses what happens to TinyML’s footprint if these systems are scaled to the number of MCUs deployed globally, which currently sits at around 250 billion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' While including all 250 billion existing MCUs in TinyML’s global footprint may not be necessary, as likely only a fraction of them will run TinyML, this aims to provide an upper bound for the net effect of TinyML both now and into the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' To continue this upper bound assump- tion, the “High-cost Profile” data from Figure 4 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This scenario results in a combined, non-trivial global carbon footprint of 1765 million metric tons of CO2-eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 1765 million metric tons of CO2-eq is a quite concern- ing footprint for TinyML on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, along with this number we need to consider the emissions that were avoided by using these systems to fairly evaluate the com- plete environmental impact of TinyML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' As mentioned earlier, there are existing examples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', [1, 22]) of simple, intelligent IoT devices which can reduce building CO2 emissions by at least 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' In this case, enduring the footprint of these smart devices is worthwhile as their footprint is most likely neg- ligible when compared to the 20% reduction in a building’s emissions they provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Figure 5 now compares the calculated global carbon foot- print of TinyML (blue bar) in the context of the emissions these TinyML systems could help avoid through efficiency improvements in other sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' If the aforementioned 20% 6 10x Other (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', Product Use, End of Life) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='00 ML Training 100 Processing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', MCU, Memory) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='00 Casing, PCB, and User Interface Transportation 5x 10x 38x 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='00 Sensing Module 10 Power Supply (3 years) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='00 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='1 16-inch Apple TinyML TinyML TinyML Medium- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='00 MacBook Watch High- Low- TinyML TinyML TinyML Pro Series 7 Cost Cost Cost High-Cost Medium-Cost Low-Cost 349 34 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='9Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' If all 250 billion MCUs were TinyML systems with three-year lifespans, their worst-case footprint would be 1765 million metric tons of CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Fortunately, if these systems enabled a 20% reduction in emissions for the residential sector and only a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='6% reduction in all other sectors (Figure 2), the total footprint would be net-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Anything larger, 20% shown as an example, results in more carbon savings from TinyML than emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' reduction in a single building’s emissions were applied to the entire residential home sector over three years (green bar), 1181 million metric tons of CO2-eq would be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' These avoided emissions alone would offset 67% of the worst- case costs of TinyML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The residential sector, though, only represents 6% of total global emissions (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' If remain- ing TinyML devices were able to reduce emissions from all other sectors by as little as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='6% on average (orange bar), then TinyML would break even from an emissions standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Furthermore, if we were to extrapolate this 20% reduction in the residential sector to all sectors (yellow bar) we would see a net reduction in global CO2 emissions by over 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='4 billion metric tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' While a 20% reduction in emissions from all sectors may be unrealistic, anything greater than a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='6% reduction on average would result in TinyML saving more emissions than it produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This result suggests that TinyML systems, if designed with careful intention, can elicit an overall positive impact on the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='5 Limitations of Our Study In this section, we recognize the limitations of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' While our research provides a starting point, more needs to be done to fully understand the impact of TinyML on environmental sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' One major limitation is the lack of publicly available data on the environmental impact of modern digital electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This makes it difficult for our analysis to be detailed and pre- cise, and also makes it challenging for consumers to make informed decisions about the environmental impact of their purchases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, it is promising to see that there is in- creasing demand for LCA and carbon footprint data in the information and communications technology industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This additional data will empower consumers and make it feasible to comprehensively account for the heterogeneity in TinyML and IoT systems in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Another important consideration is Jevons paradox, which suggests that advancements in efficiency can lead to an over- all increase in consumption and a negative impact on the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='4 attempts to address this by exam- ining a scenario in which TinyML systems are produced at a large scale (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', 250 billion TinyML systems) to improve the efficiency and sustainability of other sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This anal- ysis suggests that TinyML systems have the potential to have a positive impact on the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, the conclusions of this assessment are limited by the current data limitations, and more research is needed to make more accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 5 Future Sustainable TinyML While TinyML has the potential to contribute to global sus- tainable development and environmental sustainability, there are still many challenges that must be overcome to fully real- ize this potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' As we have shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='4, the envi- ronmental impact of TinyML will be non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Even if the benefits of TinyML can potentially outweigh its impact, it is important to be cautious and ensure that future genera- tions of TinyML devices are sustainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' In this section, we will discuss the broader implications of our study and suggest ways to make TinyML more sustainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' It should be noted that for sustainable design to be truly successful, though, incentives must be provided to corporations and engineers to prioritize sustainability when making decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Energy Harvesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Our analysis in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='3 revealed that the batteries used to power TinyML devices dominate their environmental impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Moreover, batteries present sev- eral other environmental issues, such as water and air pollu- tion and the release of carcinogens [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Research in energy harvesting [28] could reduce the batteries needed and the associated environmental costs that come with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Further- more, advancements in intermittent computing [12] could 7 2000 1765 (1181) 1000 (584) (18,991) 0 Total TinyML 20% Savings in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='6% Savings in all 20% Savings in all Footprint Residential Sector Other Sectors Other Sectors 1000 High-Cost (3 Year) (3 Years) (3 Years) 2000 18000 19000be suitable in TinyML scenarios, which would further reduce the needed power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Efficient Sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' The second largest contributor to the footprint in two of the three profiles analyzed was the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' In these two cases, a camera sensor was used, whereas in the other case, a microphone was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Sensing is essential in TinyML, but using smaller (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', camera vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' inertial mea- surement unit) or lower-quality sensors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', low- vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' high- resolution camera) could further reduce the environmental impact of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Due to the relatively low footprint of the compute, more advanced TinyML models could be used to make up for the loss in performance introduced by lower-quality sensors, thus potentially reducing the overall footprint while achieving the same performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Addition- ally, sensor fusion, which uses multiple small sensors to make up for the functionality of one large sensor, could also be used to reduce the environmental impact of sensing [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Datasheets for ML Sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Greater transparency re- garding the system’s data and costs is needed to deploy these TinyML devices safely and ethically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' TinyML instantia- tions must clearly and transparently articulate their privacy and security boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' One solution to address privacy concerns is to separate the input sensor data and ML pro- cessing from the rest of the system at the hardware level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Also, new supplementary information is needed in the form of a datasheet that builds upon traditional datasheets used for electrical components to enable transparency to end users [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' These datasheets should include information about the environmental impact and LCA of the device in an easy-to-understand format so that users can use TinyML devices in an environmentally friendly and sustainable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Datasets for Low-Resource Domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Many TinyML applications involve embedded sensors and require real- world data, which can be difficult to find, especially in public domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' There is a need for large, open-access datasets that focus on low-resource, high-impact problems involving sen- sors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Thus, to support TinyML development, it is necessary to create a dataset similar to ImageNet for TinyML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Just as open-source software development has allowed the industry to share code and reduce development costs, a shift is needed in thinking about sharing real data to create large, public, and representative datasets that can support TinyML use cases and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Furthermore, while collecting data for TinyML is vitally important, it is also important to consider the environmental impact of the data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' For example, collecting data in nature can be disruptive and harmful to habitats as data may only be available in remote locations that require travel (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=', boats or aircraft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' On the other hand, alliterative methods such as computationally-intensive simulations may require carbon-consuming resources to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Some of these costs may have to be incurred to provide a larger benefit to these environments in the long term, but these costs should be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Data sharing needs to be encouraged so that collection has to happen as few times as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Emerging Technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' New technologies are being de- veloped that could lead to more sustainable TinyML practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' One example is flexible electronics, which has demonstrated a 1000× smaller carbon footprint compared to traditional silicon manufacturing [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Such technologies could enable TinyML to achieve greater reductions in emissions than an- ticipated in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, these technologies are not yet mature and have less processing power than tradi- tional silicon devices [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' More research and development is needed to utilize these sustainable technologies fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Recycle and Upcycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' TinyML can potentially exacer- bate the problem of electronic waste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, recycling and reusing TinyML devices is a viable option as many of the algorithms can run on standard, commonly used MCU hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This can extend the life of the MCU and reduce the amount of waste sent to landfills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' It’s worth noting that in our analysis of TinyML systems in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='3, a three- year lifetime was assumed as a fair comparison with other LCA data from Apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, in reality, TinyML systems can often last closer to 10 years, which would further reduce their environmental impact over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Finally, for TinyML to have a significant impact on a global scale, it takes more than just technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' There is a need for global access to hardware and educational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Fortunately, there have been some recent efforts, led by the TinyML foundation3 and the TinyML Open Educa- tion Initiative (TinyMLedu)4, among others, to both develop such open-source materials and provide low-cost or no-cost hardware to learners [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 6 Conclusion Using ML on microcontrollers can have a significant impact on environmental sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Low-power ML on low-cost MCU-class hardware has the potential to improve efficiency in various sectors, enabling significant reductions in carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This assessment shows that TinyML’s carbon footprint could be offset by using the technology to reduce emissions from other economic sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' However, TinyML’s footprint is not negligible when scaled globally, and thus designers must be mindful and factor in sustainability when developing new devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' Emerging technologies may further enable more sustainable computing practices and cement the net-positive potential of TinyML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 7 Acknowledgements We thank Carole-Jean Wu, Udit Gupta, David Brooks, Danilo Pau, and Paul Whatmough for insightful discussions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' This work would not have been possible without their guidance and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content=' 3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='tinyml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='org/ 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} +page_content='tinymledu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfwi40/content/2301.11899v1.pdf'} 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a/jdFMT4oBgHgl3EQf5zEl/vector_store/index.faiss b/jdFMT4oBgHgl3EQf5zEl/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..a5517ea86e303b23c44cfbc7a6f724207341959d --- /dev/null +++ b/jdFMT4oBgHgl3EQf5zEl/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:534cf7efca10e6c158e67599b762377f33952a6ae4e699c5bcd1a8491db94af1 +size 4194349 diff --git a/jtE2T4oBgHgl3EQfIQaF/content/tmp_files/2301.03678v1.pdf.txt b/jtE2T4oBgHgl3EQfIQaF/content/tmp_files/2301.03678v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..500d43306b6957d25e70bbaf80abb2d95d2e9760 --- /dev/null +++ b/jtE2T4oBgHgl3EQfIQaF/content/tmp_files/2301.03678v1.pdf.txt @@ -0,0 +1,1496 @@ +Numerical simulation of the radiation force from transient acoustic fields: +Application to laser-guided acoustic tweezers +Shuhan Chen, Qing Wang, Qi Wang, Jia Zhou,∗ and Antoine Riaud† +State Key Laboratory of ASIC and System, School of Microelectronics, Fudan University, Shanghai 200433, China +(Dated: January 11, 2023) +Using pulsed acoustic waves could provide a superior selectivity for microscale acoustic tweezers. +However, the theory for the radiation force of pulsed acoustic waves has only been recently derived +and no numerical implementations are available. In this paper, we present a finite-element imple- +mentation of this model to simulate the transient acoustic radiation force on small spheres. We use +the model to simulate laser-guided acoustic tweezers and optimize their performance. By enabling +numerical simulations of the transient radiation force, this work may accelerate the rational design +of pulse-based high-selectivity acoustic tweezers devices. +I. +INTRODUCTION +The acoustic radiation force (ARF), a steady force +created by large-amplitude acoustic waves, is a conve- +nient means to achieve micro-object manipulation such +as micro-sample separation [1–3] and enrichment [4], cell +sorting [5, 6], and single-cell manipulation [7]. +Using transient excitation such as pulses could enable +much more precise manipulation than when using time- +periodic acoustic fields [1–7]. First, pulsed acoustic ma- +nipulation is less disturbed by Rayleigh acoustic stream- +ing [8, 9] because radiation force is established much +faster than streaming [10, 11]. +Second, using acoustic +wave packets allows to localize the acoustic interference +pattern and therefore to control the spatial extent of the +acoustic trapping region [12]. +Indeed, standing waves +exert radiation forces considerably larger than traveling +waves (in the small particle limit), which allows to ne- +glect the acoustic field outside the interference region. +Laser-guided acoustic tweezers (LGAT) [13] use this in- +terference principle to create a hybridized radiation force +landscape that couples a high-amplitude piezo-generated +acoustic field (strong, Z-field) acoustic field and a light- +patterned photogenerated acoustic field (weak, L-field). +The hybridized field retains the spatial information of the +L-field and the strength of the Z-field. +Despite these potential applications, theoretical and +numerical studies of transient acoustic fields are still rare. +A crying example of the limited current understanding of +transient nonlinear acoustics is the suppression of acous- +tic streaming by acoustic pulses [8, 9], where the only +model available for transient streaming [14] has been un- +able to qualitatively explain experimental observations +[10, 11]. +Likewise, there are no numerical schemes to +study transient ARF directly. +In this paper, we implement the recent generalization +of the radiation force theory for small spheres (Gor’kov +theory [15]) to transient acoustic fields [16]. Besides re- +quiring that the objects are spherical and much smaller +∗ jia.zhou@fudan.edu.cn +† antoine riaud@fudan.edu.cn +than the acoustic wavelength at the considered band- +width, this theory overlooks micro-streaming and would +not be suitable for studying very dense particles in highly +viscous fluids (such as copper in glycerol [17]). It also +requires that scattering events do not overlap and there- +fore extreme care must be taken when investigating the +response of bubbles or other high-quality acoustic res- +onators. Nonetheless, the underpinning assumptions sug- +gest that the theory is valid for cells or plastic particles +immersed in water. +The manuscript is arranged as follows. After summa- +rizing the theory of dynamic acoustic radiation force, sec- +tion II details a numerical implementation of the dynamic +ARF and details the model setup for the simulation of an +LGAT. Section III briefly presents the acoustic field in an +LGAT device and then analyzes the resulting ARF de- +pending on the location of the particle and on the phase +difference between the laser and the piezoelectric. Then, +the excitation parameters of the LGAT (pulse duration +and laser beam width) are optimized to maximize the +ARF. +II. +THEORIES AND METHODS +A. +Transient ARF: Governing equations +Our simulations implement the theory of ARF on small +rigid spheres in transient acoustic fields of Wang et al. +[16]. +Similar to most other theories of radiation pres- +sure, the wave peak pressure amplitude pmax is assumed +to be small such that pmax/ρ0c2 +0 = ϵ ≪ 1 where ϵ is the +acoustic Mach number, with ρ0 the quiescent fluid den- +sity. +According to the perturbation method, the fluid +quantities can be resolved as +p = p0 + p1 + p2, +(1a) +v = v0 + v1 + v2, +(1b) +ρ = ρ0 + ρ1 + ρ2, +(1c) +where subscripts 0 − 2 indicate the perturbation order of +each term q0 = q1O(ϵ) = q2O +� +ϵ2� +where q can be any +physical field among pressure, density and velocity. Ac- +cordingly, time-invariant 0-order terms are interpreted as +arXiv:2301.03678v1 [physics.flu-dyn] 9 Jan 2023 + +2 +constant (hydrostatic) contributions, 1st order terms as +acoustic contributions, and 2nd order terms as nonlinear +effects such as radiation pressure or acoustic streaming. +Although the theory of radiation force requires knowl- +edge of the total acoustic field, Gor’kov theory is conve- +niently expressed using the incident acoustic field only. +Henceforth, all acoustic quantities will refer to the inci- +dent field in the remaining of the paper. +Compared to Gor’kov theory, the assumption of single- +frequency acoustic fields is extended to wave packets of +finite duration τ: all the wave quantities q1 are required +to keep the initial state and final state consistent, which +means satisfying the condition q1 (0, r) = q1 (τ, r), where +r denotes the position vector and the origin of time can +be chosen arbitrarily. Additionally, pulse duration must +be short enough to neglect the sphere displacement com- +pared to the shortest wavelength, which can be evaluated +by 1/τ ≫ fmaxϵ2, where fmax refers to the highest fre- +quency in the pulse. +In the case of an inviscid fluid (viscoacoustic and ther- +moacoustic boundary layers much thinner than the parti- +cle diameter [18, 19]), the acoustic radiation force reads: +Frad = −Vp∇U, +(2) +with the particle volume Vp = 4πR3 +p/3 and the general- +ized Gor’kov potential: +U = ⟨Uinst⟩ = +1 +ttot +� +Uinst dt, +(3) +where Uinst is interpreted as a fictive (but convenient) +instantaneous Gor’kov potential: +Uinst = f1V − 3 +2f2K. +(4) +Here K = ρ0v1 · v1/2 and V = p2 +1/(2ρ0c2 +0) represent in- +stantaneous kinetic energy and potential energy respec- +tively. f1 = 1 − ρ0c0/(ρpcp) is the monopole scattering +coefficient and f2 = 2 (ρp + ρ0) /(2ρp + ρ0) is the dipole +scattering coefficient. +Therefore, computation of the radiation force requires +knowing the incident acoustic field, which can be com- +puted using transient acoustic pressure models: +1 +c2 +0 +∂2p1 +∂t2 + ∇2p1 = 0, +(5) +from which the fluid acceleration can be deduced using +the Euler equation: +a1 = − 1 +ρ0 +∇p1, +(6) +and the velocity: +v1 = +� +a1 dt. +(7) +Under these hypotheses, the general Gor’kov approach +can be used for calculating ARF in such transient situa- +tions. +B. +Transient ARF: Numerical implementation +The numerical model for the simulation of hybridized +acoustic fields was configured in the commercial finite- +element-method software COMSOL Multiphysics® ver- +sion 5.4, and the parameters used are given in Ta- +ble I. The model uses the “Pressure Acoustics, Transient” +module to solve the first-order acoustic pressure field in +the fluid domain (Eq. (1b) and to deduce the accelera- +tion and velocity fields with Eq. (7)). The settings for +transient wave solving follow the approach suggested by +Ref. [20]. +Transient acoustic field studies are necessarily associ- +ated with bandwidth issues. On the one hand, the exci- +tation signal (such as a short laser pulse) may have a very +wide bandwidth. On the other hand, integration of the +acoustic partial differential equation is often done with a +constant time step and the grid imposes constraints on +the minimum wavelength that can be resolved (that is, +the maximum frequency). The Courant-Friedrichs-Lewy +(CFL) condition provides guidelines to set the time step +∆t depending on the maximum mesh size h [14, 20]: +CFL = c∆t +h , +(8) +where c is the wave speed. +Using the default second- +order (quadratic) mesh elements, the recommended CFL +number is approximately 0.1. Thus, given a maximum +frequency fmax that needs to be resolved, the CFL yields +the time step ∆t = CFL/(Nfmax) where N is the number +of elements per local wavelength λ. Due to the period +doubling of the Gor’kov potential (terms in p2 +1 and v2 +1), +we choose N = 12 instead of the more common N = 6 in +linear acoustics. +Boundary conditions must be designed carefully ac- +cording to this bandwidth constraint. The most straight- +forward is to directly set an acceleration that fits in the +simulated frequency bandwidth as the boundary condi- +tion. However, in some cases, the velocity needs to be +specified instead (for instance when the velocity is known +from laser Doppler vibrometer measurements) which can +yield to step-wise acceleration if there is a slope-break in +the velocity. To address this problem, the input veloc- +ity is first multiplied by a smoothing window, and then +derived to obtain the acceleration boundary condition. +The velocity field is computed (Eq. (7)) by coupling the +pressure acoustic model to a “Distributed Ordinary Dif- +ferential Equation (DODE)” module in COMSOL with +the damping factor set to 1 in DODE. +The first-order velocity is then used to calculate the +instantaneous Gor’kov potential Uinst via Eq. (4) along +with the first-order acoustic pressure. Similarly, Eq. (3) +is implemented in the “Distributed Ordinary Differen- +tial Equation (DODE)” module to obtain U. +Finally, +the ARF is obtained as the gradient of U (using the +COMSOL function diff()). Given the small size of the +geometry, computer memory is not a constraint in this +study, and all the fields are computed using a single fully- + +3 +coupled time-dependent solver for simplicity, but the per- +turbation approach of the model could allow using a seg- +regated solver that solves the fields one after another for +each time step. +TABLE I. Summary of simulation parameters. +Electroacoustic wave parameters +Frequency +f Z +10 MHz +Velocity magnitude +vZ +0 +1.83 m/s +Number of shots +nZ +5 +Density +ρz +4640 kg/m3 +Laser-induced acoustic wave parameters +Frequency +f L +10 MHz +Velocity magnitude +vL +0 +0.6 m/s +Radius of spot waist +Rw +20 µm +Fluid +Density +ρ0 +1050 kg/m3 +Speed of sound +c0 +1500 m/s +Domain width (radius) +Rd +375 µm +Domain height +hd +200 µm +Particle +Density +ρp +1050 kg/m3 +Radius +Rp +10 µm +Mesh & Solver +CFL number +CFL +0.5 +Units per local wavelength +N +12 +Others +Simulation time +tsim +2 µs +Duration between pulse series +ttot +20 µs +C. +Simulation of an LGAT: Initial and boundary +conditions +In an LGAT (schematic shown in Fig. 1), two acoustic +sources interfere to generate an acoustic trap. A piezo- +electric acoustic source (strong field, labeled with the +superscript “Z”) yields a plane traveling wave of high +amplitude which creates no in-plane acoustic radiation +force due to symmetry constraints (the wave is in-plane +invariant). This wave interferes with a weaker acoustic +wave (weak field, labeled with the superscript “L”) gen- +erated by the photoacoustic conversion of a laser pulse. +The combined incident pressure and velocity fields read +p1 = pL +1 + pZ +1 and v = vL +1 + vZ +1 , respectively. Accord- +ing to Eqs. (3) and (4), this yields a Gor’kov potential +with cross terms U = U ZZ + U ZL + U LL. The vanishing +gradient of spatially invariant U ZZ leads to a negligible +in-plane force, while U LL is negligible due to the small +photoacoustic conversion efficiency. As a result, only the +hybridized potential U ZL dominates, and the force is well +approximated by: +Frad = −Vp∇U ≈ −Vp∇U ZL = −Vp∇ +� +U ZL +inst +� +, +(9) +with U ZL +inst = f1pZ +1pL +1/(ρ0c2 +0)− 3 +2f2ρ0vZ +1 ·vL +1 . We note that +Uinst is twice larger than in Eq. (4) due to the binomial +coefficients in the expansion of p2 +1 and v1 · v1. +FIG. 1. +Laser-guided acoustic tweezers (LGAT) structure. +The particle is manipulated by the combination of a weak and +a strong acoustic field. The interference between the two fields +yields a radiation force with spatial characteristics similar to +the weak field but with a much higher amplitude. The L-field +is generated by the photoacoustic conversion of a pulsed laser +beam on the gold nanoparticle-polydimethylsiloxane (AuNP- +PDMS) composite. The composite selectively absorbs green +light and is transparent to other wavelengths. +The Z-field +is generated by piezoelectric conversion of an electric signal +(alternating current, a.c.) +in the LiNbO3. +The electrodes +(indium tin oxide, ITO) are transparent. +As a result, the +whole device is transparent to most optical wavelengths ex- +cept green, and the particles under manipulation can be di- +rectly visualized. +While we are mainly focused on the acoustic radiation +pressure, we note that acoustic streaming plays a negligi- +ble role in this study due to the particle radius of 25 µm +used in the experiments [21]. For fluids in the channel, +thermophoresis and thermal agitation can also be factors +driving the particles [22]. To rule out this possibility, we +have turned off the bulk acoustic wave (BAW) from Z- +source and observed no motion of particles despite the +laser heating [13]. Therefore, thermal effects are also ne- +glected in the model. +The other parameter values are +given in Table I. Most of these parameters are selected +based on the typical experimental values while they are +sometimes simplified to facilitate dimensionless analysis +(for instance f Z is rounded to 10 MHz instead of the exact +resonance frequency of the crystal used in experiments +(7.6 MHz)). Further, an important goal of the paper is +to estimate what would be the upper limit of the radi- +ation force of the LGAT if suitable laser and electronic +excitation were provided. For the electroacoustic field, a +simple estimate is obtained by using the definition of the +piezoelectric coupling coefficient K2: +1 +2C(Ed)2K2 ≃ 1 +2ρzAdvZ +0 +2, +(10) +where d is the thickness and A is the area of the dielectric +crystal, together with the capacitance C = (εrε0A)/d +and the field strength E. On this basis, an estimation +of the velocity is derived as vZ +0 = KE +� +εrε0/ρz. This +velocity is independent of the excitation frequency and +is limited by the coercivity field strength of the crystal, +which is of the order of 21 MV/m for LiNbO3 [23]. The +Y-36° cut has a K2 = 0.1. +For the sake of simplicity, +the material anisotropy is overlooked and we consider a + +Chamber +particle +water +AuNP-PDMS +PDMS +ITO +a.C +LiNbO +ITO +laser4 +FIG. 2. +Model setup. +(a) Sketch of the two-dimensional +axial symmetry computational domain. The purple domain +is used for the simulation of the Z-field and the calculation of +the radiation force. The gray domain is used for the calcula- +tion of the L-field. (b) Input signals applied to the bottom +boundary. +Input signals are depicted by velocities varying +with time. Grey and green curves represent Z-signal (elec- +troacoustic BAW) and L-signal (photoacoustic pulse) respec- +tively. The zoom-in shows the signal duration overlap under +typical relative phases (∆ϕ = ±π/2) within the time window +of a period (from 0.47 µs to 0.62 µs ). +relative permittivity of the order of εr ≈ 40, which yields +a maximum velocity of 1.83 m/s. +To satisfy U LL ≪ U ZL as required by the hybridized- +fields theory, the maximum excitation velocity of the pho- +toacoustic field is set to 60 cm/s. We note that the ma- +terial damage threshold could in principle allow higher +velocities. More details about the criterion are given in +Appendix A. +The laser spot being axisymmetric and the Z-wave be- +ing spatially-invariant, we use a two-dimensional axisym- +metric model (cylindrical coordinates system) as shown +in Fig. 2(a). We restrict our analysis to a rectangular +cross-section (purple part in Fig. 2(a)) of width Rd = +nλλZ/2 = 375 µm (with nλ = 5) and height hd = 200 µm +in the vertical r-z plane for a 10 MHz wave. The width +is chosen large enough to ensure that the tweezers yield +a negligible force at that distance. +In experiments, the particles are contained in a poly- +dimethylsiloxane (PDMS) microchannel filled with wa- +ter. PDMS and water have a similar acoustic impedance +(plane wave reflection coefficient R = 18%). +In order +to reduce simulation time, the simulation domain size +is minimized by neglecting acoustic reflections on the +PDMS. Instead, plane wave radiation boundary condi- +tions are used for the Z-wave, and spherical wave radi- +ation ones are used for the L-wave. A virtual boundary +of radius Rvb = +� +R2 +d + h2 +d + λZ/2 shown in Fig. 2(a) +is constructed to facilitate the setting of spherical wave +radiation boundary condition. +More refined boundary +conditions accounting for reflections in the PDMS-water +surface and shear waves in the PDMS are discussed by +Ni et al. [24]. +The transducer is not explicitly modeled, instead, +the piezoelectric Z-excitation is implemented as an r- +invariant acceleration: +aZ = aZ +0sin +� +ωZt − ϕZ� +Λ +�ωZt − ϕZ +2π +� +. +(11) +The triangular window Λ(·) with the trigger range of +[0, nZ] simulates the resonance onset of the piezoelec- +tric with a train of nZ = 5 cycles, (see Fig. 2(b)). We +note that longer excitation could be possible but would +be more sensitive to reflections at the lateral edge of +the wafer that would induce standing waves (and violate +the in-plane invariance of the Z-wave). The relationship +ωL = ωZ exists unless specified otherwise. +Likewise, the photoacoustic excitation is described by +an acceleration with a Gaussian intensity profile, since +the photoacoustic vibration is well approximated by a +Gaussian profile according to our experimental measure- +ments [25]. We set the radius of the beam waist Rw close +to the experimental value (20 µm), producing a normal +acceleration given in Eq. (12), where the amplitude reads +aL +0 = vL +0 /ωL . +A rectangular window function modu- +lates the sinusoidal signal to simulate the experimentally- +detected pulsed L-signal (see Fig. 2(b)): +aL = ∂vL +0 +∂t += aL +0e−(r/Rw)2 ∂ +∂t +� +sin +� +ωLt − ϕL� +Π +�ωLt − ϕL +2π +�� +. +(12) + +(a) +BC: spherical +wave radiation +BC: plane wave radiation +2/2 +BC: symmetry +R +fluid:domain +Rd +BC: normal acceleration +(b) +×108 +0.8 +Z +0.6 +0.4 +0.2 +(m/ +0 +a +-0.2 +-0.4 +-0.6 +-0.8 +0 +2 +t (us) +元5 +Here the rectangular window Π(·) is set to 1 over the +interval [0, 1/2] to simulate a pulse of duration τ L = π/ωL +(here τ L = 50 ns). +A smoothing of 0.4 is introduced to introduce frequen- +cies above fmax = f Z that would trigger numerical insta- +bilities (see Appendix B). The smoothing process intro- +duces small overshoots at the beginning and the end of +the laser waveform function with an amplitude negligible +compared to the main pulse peak (≈ 20%). Both excita- +tions have an adjustable phase ϕ that can be set in exper- +iments using an adjustable delay to a common reference +trigger. By default, ϕZ = 0 and ϕL = (2nZ + 1/2)π such +that the two excitation waveforms are in-phase. When +the phase between Z and L is adjusted, ϕZ is the adjust- +ment variable (ϕL remains unchanged) and an empirical +reference ϕ0(r) is introduced such that ∆ϕ = ϕZ − ϕ0 +vanishes when the force is 0 at the measurement point r. +This will be discussed in more detail in section III B 3. +III. +RESULTS +In this section, we illustrate the working of the laser- +guided acoustic tweezers and discuss potential directions +for parameter optimization. +A. +Acoustic field +As shown in Fig. 2(b), the simulation starts at t = +−0.2tsim = −0.40 µs. At t = 0, the piezoelectric begins +to emit a BAW (Z) wave. The photoacoustic pulse (L- +signal) hits the substrate at t = 0.52 µs. Therefore, the +observation of time evolution begins at t = 0.52 µs. +FIG. 3. +Acoustic pressure of the L-field on the central axis +of the simulation domain. As shown in Fig. 2, the laser beam +hits the substrate at t = 0.52 µs. +The Z-field is a plane traveling wave and remains sim- +ilar to Fig. 2(b) as it travels in the z-direction. The L- +wave generated by photoacoustic conversion is shown in +Fig. 3. It is essentially a spherical wave that behaves sim- +ilarly in the z- and r-directions (r-profile available in Ap- +pendix C). The L-acoustic pressure reaches its maximum +amplitude immediately after generation at t = 0.54 µs +and decays by 80% by the time it exists in the simula- +tion domain (t = 0.66 µs). +B. +Acoustic radiation pressure +In manipulation experiments, the particle is confined +in the manipulation chamber. +Therefore, radial forces +tend to be more important than axial ones. Hence, un- +less specified otherwise, the ARF will refer to F ZL +r += +−Vp ∂U ZL +∂r . +1. +Build-up of the Gor’kov potential +Fig. 4 shows the build-up of the acoustic radiation pres- +sure. The pressure of the Z and L-waves are shown in +Fig. 4(a): the Z-wave is a traveling plane wave, and the +L-wave is similar to a spherical wave. +Despite being a virtual quantity, the instantaneous po- +tential Uinst to visualize the hybridization of the L and +Z fields during the acoustic radiation pressure build-up. +For example, the region where Uinst < 0 in Fig. 4(b) cor- +responds to the mixing of pL +1 and pZ +1 of opposite signs. +This instantaneous Gor’kov potential grows over time, +as shown in Fig. 4(c). +As the spherical wavefront (L) +propagates, the inhomogeneous Gor’kov potential region +spreads to the entire simulation domain. A cross-section +of the hybridized Gor’kov potential U ZL experienced by +a particle resting on the bottom wall (z = Rp) is shown +in Fig. 4(d). The potential is mainly concentrated at the +location of the laser spot, which ensures good selectivity +for acoustic trapping and manipulation. We note that +our model neglects the effect of walls, which needs to be +accounted for using a more advanced theory (see Ref. [26] +for the monofrequency case). +2. +Effect of the particle location +The effect of the particle location is shown in Fig. 5, +with ∆ϕ = π/2 selected to maximize the radiation force +(this will be interpreted in section III B 3). The closer +the observation position z is taken to the lower edge, the +larger the ARF peak generated. The concentration of the +pulse energy is also reflected in the ARF curves shown in +Fig. 5(a) and (b), where the ARF peaks show a decay- +ing trend when moving away from the excitation source +from either the r-direction or the z-direction. According +to Fig. 5, not only the force becomes very weak when the +particle is located near the top of the channel, but the +selectivity (ratio of primary force maximum to the sec- +ondary one) also deteriorates. It is therefore important +in LGAT design to ensure that the particles are relatively + +×105 +(us) +0.66 +a +4 +3 +0.64 +Acoustic Pressure +2 +0.62 +1 +0.6 +0 +-1 +0.58 +-2 +0.56 +-3 +-4 +0.54 +-5 +0.52 +0 +50 +100 +150 +200 +z-coordinate (um)6 +FIG. 4. +Time-dependent field evolution with ∆ϕ = 0. Pictures are taken at t = 0.54, 0.62, 0.70, and 0.78 µs. (a) Super-imposed +acoustic pressure field. The blue-red color represents the L-field pressure and the contour represents the Z-field pressure. (b) +Instantaneous Gor’kov potential Uinst. (c) Time-averaged Gor’kov potential U. (d) Cross-section of the magnitude of U at +z = Rp. +FIG. 5. +Variation of the hybridized acoustic radiation force +F ZL +r +at various heights in the manipulation chamber when +∆ϕ = π/2. +close to the laser spot, which may limit the biocompati- +bility of the manipulation, unless holographic techniques +are used [27]. In the following, the force is maximized +by assuming that the particle rests at the bottom of the +channel (z = Rp) +3. +Effect of the phase difference between Z and L fields +A distinctive feature of LGAT is the possibility to com- +mute between attractive and repelling forces by adjust- +ing the electronic delay between the triggers of the laser +source and the electric signal generator. In the simula- +tion, this is implemented by changing the relative phase +difference ∆ϕ between the Z and L fields, as shown for +a particle located at z = Rp = 10 µm, r = 3Rp = 30 µm +(Fig. 6). Following a convention proposed in our previ- +ous paper, we add an offset ϕ0(r) to ∆ϕ such that the +ARF cancels for ∆ϕ = 0 [13]. A better alternative will +be proposed later on. + +(a) +t =0.54μs +t = 0.70 μs +t = 0.78 μs +ApL Apz +t = 0.62 μs +Pa +Pa +Z-coordinate ( +×105 +×106 +100 +1.18 +0.24 +10 +-0.7 +-1.64 +0 +100 +200 +300 +r-coordinate (um) +(b) +200 +t = 0.54 μs +t = 0.62μs +t = 0.78μs +Uinst +t = 0.70 μs +J/m3 +z-coordinate +200 +100 +1-200 +0 +100 +200 +300 +r-coordinate (um) +(c) +(um) +200 +t = 0.54 μs +t = 0.62 μs +t = 0.70 μs +t = 0.78 μs +U +J/m3 +z-coordinate ( +0.2 +100 +0 +-0.2 +0 +100 +200 +300 +r-coordinate (um) +(d) +t = 0.54 μs +t = 0.62μs +t = 0.70 μs +t =0.78 us +J/m3 +-0.1 +-0.05 +0 +0.05 +0.1×10-12 +0 +-5 +-10 +-15 +(N) +-20 +-25 +-30 +-35 +2R, +-40 +R +-45 +R,/2 +-50 +0 +100 +200 +300 +r-coordinate (um)7 +FIG. 6. +Hybridized acoustic radiation force F ZL +r +at z = Rp = +10 µm, r = 3Rp = 30 µm depending the phase difference ∆ϕ +between the Z and L fields. +The variation of the ARF at fixed z = Rp = 10 µm +but with a varying phase is shown in Fig. 7. Here, the +phase offset is kept at 0 such that ∆ϕ = ϕZ every- +where. Although at r = 3Rp = 30 µm the force behaves +consistently with Fig. 6 (strongest pull at ∆ϕ = π/2, +strongest push at ∆ϕ = −π/2 and vanishingly small at +∆ϕ = 0, it can be seen that points at half a wavelength +away (such as r = 105 µm) behave in the opposite fash- +ion and points at a quarter wavelength distance (such as +r = 62 µm) behave in phase quadrature (strongest pull +at ∆ϕ = π, strongest push at ∆ϕ = 0 and vanishingly +small at ∆ϕ = ±π/2). This emphasizes that the exact +delay between L and Z fields depends not only on the +instruments trigger but also on the acoustic propagation +of both fields. The latter is relative to the exact point +in space where the ARF is measured. This issue is ad- +dressed in the next section. We also note that the force is +not exactly periodic over time due to the slow variations +of the wave envelope (the wave amplitude is not the same +for ∆ϕ = −π and ∆ϕ = π). +4. +A phase-independent factorization of the hybridized +Gor’kov potential +In the previous section, we have shown that the phase +difference ∆ϕ is an important factor controlling the ARF +of LGAT, but that the relationship between the force di- +rection and ∆ϕ depends on the spatial location of the +particle being manipulated. +Hence, gaining a perfect +knowledge of the ARF of an LGAT would require re- +peating ARF calculations for all phases ∆ϕ ∈ [−π, π]. +Here, we show how to reduce this large number of cal- +culations to only two. Ref. [13] gives a way to obtain +the hybridized potential U ZL by setting up an analyti- +cal Z-field wave with constant amplitude. Then U ZL is +expressed as a linear combination of two basis potentials +FIG. 7. +Variation of the hybridized acoustic radiation force +F ZL +r +at z = Rp depending on the phase difference ∆ϕ be- +tween the Z and L fields. The line color indicates the phase +while the line width grows with the phase in order to visually +distinguish between ∆ϕ = −π and ∆ϕ = π. +weighted by trigonometric functions of ∆ϕ: +ˆU ZL = Φccos ∆ϕ + Φssin ∆ϕ, +(13) +with the basis potentials Φc (r) = +� +pZ +1|∆ϕ=0EL� +and +Φs (r) = +� +pZ +1|∆ϕ=π/2EL� +independent of ∆ϕ, where L- +component EL = f1pL +1/(ρ0c2 +0) − 3 +2f2vL +1z/c0. +Here only +the z-direction component of vL +1 is considered due to +z-propagating-only vZ +1 . +ˆU ZL obtained from Eq. (13) is +strictly valid only for monofrequency Z-wave but is as- +sumed here to be a good approximation of the actual +potential U ZL. In the simulations, Φc (Φs) is obtained +by setting ∆ϕ = 0 (∆ϕ = π/2). The approximate poten- +tial can then be computed from these two basis potentials +according to Eq. (13). +Fig. 8(a) shows the basis potentials at z = Rp. Their +values can be recorded in the entire simulation do- +main. The quality of the approximation is evaluated in +Fig. 8(b): the approximated potential matches U ZL (from +Eq. (3)) well, although the relative error for ∆ϕ = −π +increases by ∼ 8% compared to the one of ∆ϕ = π/10. +This is likely because the Z-wave has a triangular enve- +lope of varying amplitude which is not considered in the +theory of Ref. [13]. +C. +Optimization of operating parameters +The main weakness of current LGAT implementations +is that they yield a very small force that results in dis- +placement speeds of a few micrometers per second. In the +following, we optimize two parameters readily adjustable +in experiments: (1) the ratio of the L-wave frequency +(reflecting the laser pulse duration) with respect to the +Z-wave frequency, i.e., ξf = f L/f Z (which can be ad- +justed on most pulsed laser sources); (2) the size of the + +×10-12 +r = 30 um, z = 10 um +25 +20 +15 +10 +5 +(N) +0 +-5 +Z +-10 +F +-15 +-20 +-25 +-30 +-35 +-1 +0 +1 +△0 (元)△ (元) +30 +25 +0.8 +20 +15 +0.6 +10 +0.4 +505° +(N) +0.2 +0 +05252 +-0.2 +-0.4 +-0.6 +-30 +-0.8 +35 +.1 +40 +0 +100 +200 +300 +r-coordinate (um)8 +FIG. 8. +Approximate Gor’kov potential ˆU ZL with ∆ϕ decou- +pled (a) Constant basis potentials Φc and Φs. (b) Comparison +between ˆU ZL and U ZL for ∆ϕ = −π and π/10. +pulse source, expressed as the radius of the laser spot +Rb (which can be adjusted by changing the microscope +objective magnification). We assume that ξf is adjusted +by changing f L with f Z fixed at 10 MHz. In this section, +the conditions ∆ϕ = π/2 and z = Rp are retained, and +the radiation force magnitude Fr +max is discussed based +on the maximum absolute values of the ARF in the r- +direction. +To reflect experimental constraints, the comparisons +are done at constant laser peak energy (that would be +equivalent to adjusting the laser energy to avoid material +damage). +Photoacoustic generation theory shows that +the photoacoustic wave amplitude is proportional to the +laser power. Hence, we adjust the photoacoustic wave +amplitude as vL +0 ξf (see Appendix D). In addition, the +laser phase ϕL is adjusted to keep the peak aligned with +the electroacoustic field peak. +Fig. 9 illustrates the effect on ARF of modulating ξf. +The ARF increases fast as the pulse shortens until it +reaches a saturation value of Fr +max ∼ 44 pN when the +pulse frequency satisfies (f L ∼ f Z). Afterward, the in- +crease flattens. +While the marginal improvement past +f L ∼ f Z might seem attractive for improved versions of +the LGAT, we note that the short wavelength at high +frequencies violates Gor’kov’s assumption of an object +size much smaller than the wavelength, and the onset of +resonance can yield to an uncontrollable force direction. +FIG. 9. +Hybridized acoustic radiation force F ZL +r +for various +pulse duration (f L/f Z) while keeping the laser pulse energy +constant. +Next, we consider varying the laser spot size, for +instance by using objectives of varying magnification. +Here, Rb is introduced to distinguish the adjusted spot +radius from the original radius Rw used in the other +parts of the paper. The ratio ξR = Rb/Rw is defined +for convenience. To maintain constant pulse energy, the +scaling vL +0 /ξ2 +R neglecting the loss of transmission of high- +magnification objectives is used (detailed derivation in +Appendix D). +Fig. 10(a) and (b) show the evolution of the ARF de- +pending on the pulse source radius when ∆ϕ = 0 and +Fig. 10(c) and (d) shows the evolution of the ARF de- +pending on the pulse source radius when ∆ϕ = π/2. It is +found that ∆ϕ = π/2 yields the largest force regardless +of the frequency and beam radius. Optimizing the latter +two parameters (f L = 14 MHz (pulse duration of 31 ns), +Rb = 10 µm), we expect a force of ∼ 80 pN, well within +the range of other acoustic tweezers, but that would have +the extra advantage of being controlled by a light pattern. +IV. +PERSPECTIVE +In the model, we have only discussed the case of small +particles in an ideal fluid domain, and the properties of +the wave sources are relatively simplified. Therefore, the +model deserves further extensions, such as the introduc- +tion of thermoviscous effects and interaction with walls. +Furthermore, we have assumed in the model that the +radius of the particle was small enough to be consis- +tent with the Gor’kov approximation [28]. However, the +LGAT experiments [13] of manipulation of ∼ 25 µm ra- +dius particles have unveiled a much larger force, presum- +ably due to the onset of resonance. Therefore, a theory +for transient ARF valid for arbitrary particle radii would +be highly desirable. + +a +(un) +200 +Φc +(Ag = -元/2) +z-coordinate +100 +0 +100 +200 +300 +r-coordinate (um) +Φs +(A = -π) +(b) +0.1 +0.08 +0.06 +Potential (J/m3) +0.04 +0.02 +0 +-0.02 +-0.04 +(=-元) +-0.06 +.....0(A = -元) +-0.08 +U(Aβ = 元/10) +-0.1 +...0(AP = 元/10) +-0.12 +-0.14 +0 +100 +200 +300 +r-coordinate (um)×10-12 +40 +35 +30 +(N) +25 +20 +15 +10 +5 +0++ +0 +109 +FIG. 10. +Hybridized acoustic radiation force F ZL +r +for various pulse durations (expressed as frequencies of 6, 8, 10, 12, and +14 MHz) and beam radii, while keeping the beam energy constant. (a) and (b) presents the force versus the beam radius Rb +and the dimensionless radius Rb/λZ for ∆ϕ = 0 while (c) and (d) presents those for ∆ϕ = π/2. +V. +CONCLUSION +Despite the availability of a theoretical expression for +the ARF of short pulses, no finite element models had +been implemented so far. In this paper, we have shown +that such a simulation could be carried out by (i) using +time-explicit acoustic pressure simulations to compute +the acoustic field, (ii) defining a virtual instantaneous +Gor’kov potential, and (iii) integrating it over time to ob- +tain the time-averaged Gor’kov potential. We have then +used our code to simulate laser-guided acoustic tweezers. +Our simulations show that the behavior of the LGAT can +be captured by two conjugated Gor’kov potentials Φc and +Φs weighted by trigonometric functions. Optimization of +operating parameters shows that the force increases when +reducing the laser spot size, while the gain from shorten- +ing the pulse duration is limited (∼ 10%). Finally, based +on the dielectric breakdown limit of the piezoelectric, we +estimate the maximum LGAT force on 10 µm polystyrene +particles in water to ∼ 40 pN, which is comparable to +other acoustic tweezers. Beyond LGAT, the availability +of a numerical model may accelerate the development of +time-dynamic acoustic tweezers with superior dexterity +compared to their monochromatic counterpart. + +a +×10-12 +×10-12 + 6 MHz +6 MHz +24 +24 +-0- 8 MHz +.-0- 8 MHz +22 +22 +--→- 10 MHz +.→- 10 MHz +20 +20 +-- 12 MHz ++- 12 MHz +18 +18 +-+: 14 MHz +.-+- 14 MHz +16 +16 +N +14 +14 +12 +12 +F +10 +10 +8 +8 +6 +6 +4 +4 +2 +2 +0 +0 +50 +100 +0.5 +Rb (um) +Rb/aZ +(d) +C +×10-12 +×10-12 + 6 MHz +- 6 MHz +56 +-0- 8 MHz +--0- 8 MHz +-→- 10 MHz +--→- 10 MHz +385844 +- 12 MHz +-- 12 MHz +-+- 14 MHz ++- 14 MHz +Z +50 +100 +0.5 +Rb (um)10 +ACKNOWLEDGMENTS +This work was supported by the National Natural Sci- +ence Foundation of China with Grants Nos. 12004078, +61874033, and 62274039; the State Key Lab of ASIC and +System, Fudan University with Grants Nos. 2021MS001, +2021MS002, +and 2020KF006; +and the Science and +Technology Commission of Shanghai Municipality with +Grants Nos. 22QA1400900 and 22WZ2502200. +Appendix A: Magnitude comparison between U LL +and U ZL +The simulation method follows the hypothesis of the +hybridized-fields theory [13], which expects a relatively +low value of U LL compared to U ZL. U LL is independent +of ∆ϕ. Here we evaluate the effect of U LL with the max- +imum U ZL induced by adjusting ∆ϕ. +As shown in Fig. 11, the maximum U LL reaches less +than 10% of the maximum U ZL, which validates our as- +sumption for the current simulation settings. +FIG. 11. +Magnitude comparison between U LL and U ZL with +∆ϕ = ±π/2. +Appendix B: Effects of input signal smoothing +Since the acceleration is derived from the derivation +of the velocity, we will get an input aL with a sharp +variation at the beginning and the end of the signal (as +shown in Fig. 12(a)). +Large changes tend to generate +numerical errors. To minimize the issue, we introduce a +transition region of width 0.4ωL/2π on each side of the +rectangular window. +In Fig. 12(b), we check that the +deviation of the ARF resulting from the smoothing is +moderate when compared to the non-smoothed function. +FIG. 12. +Comparison between the sharp signal and the +smoothed signal. (a) Signal input. (b) F ZL +r +on z = Rp with +∆ϕ = π/2. +Appendix C: Isotropy of the laser pulse +The consistency of the variation trend and magnitude +of the acoustic field in the r-direction (Fig. 13) with that +in the z-direction (Fig. 3) reflects the spherical wave ap- +proximation condition. However, in the near-field zone, +the acoustic pressure change in the r-direction is sharper. +Such difference in propagation characteristics of different +directions originates from the normal direction of the ini- +tial acceleration, while the rapid attenuation of acoustic +fields in both directions reflects the strong concentration +of acoustic energy obtained by photoacoustic generation. +Appendix D: Scaling velocity to maintain a constant +pulse energy +When changing operating parameters in the simula- +tions, the laser pulse energy is assumed to remain con- +stant. This is achieved by scaling the L-velocity accord- +ing to the Vashy-Buckingham Pi-theorem. The velocity + +0.3 +—UZL(=-/2) +0.25 +0.2 +—UZL(△=元/2) +0.15 +-ULL +5 +0.1 +0.05 +0 +-0.05 +-0.1 +-0.15 +-0.2 +-0.25 +-0.3 +-0.35 +0 +100 +200 +300 +r-coordinate (um)(a) +×107 +4 +Smoothed +*** +* Original +3 +2 +(m/s2) +1 +0 +XXXXXXXXXXX +上 +-1 +a +-2 +* +* +-3 +0 +-4 +.5 +0.5 +0.55 +0.6 +t (us) +(b) +×10-12 +5 +0 +-5 +-10 +(N) +-15 +-20 +-25 +-30 +Smoothed +-35 +Original +-40 +0 +100 +200 +300 +r-coordinate (um)11 +FIG. 13. +Propagating properties of L-acoustic pressure vary- +ing with r on the lower edge (z = 0). The color bar from light +yellow to dark red represents the change of time from 0.52 µs +to 0.66 µs. +is assumed to read: +vL(f Lt, r/Rb) = vL +0 (ξR, ξf)g(f Lt, r/Rb), +(D1) +where g is an arbitrary integrable function that can +be obtained in our simulations by integrating Eq. (12). +The goal of the following calculation is to determine +vL +0 (ξR, ξf)/vL +0 (1, 1). +The +pulse +energy +from +the +laser +reads +E += +� +Atot +� +ttot ε dS dt, where ε is the pulse energy density. +Substituting ε = β−1vL, with β the photoacoustic con- +version coefficient (assumed to be constant for the lim- +ited frequency and power range studied here) and using +Eq. (D1), we get: +E ≈ β−1vL +0 (ξR, ξf) +� +∞ +� +∞ +g(f Lt, r/Rb) dS dt. +(D2) +The right-hand side integral is obtained by assuming that +the simulation domain is large enough to completely en- +compass the laser spot. Changing the integration vari- +ables to f Lt and r/Rb, we get: +E ≈ β−1vL +0 (ξR, ξf)Rb +2 +f L Ig, +(D3) +where Ig is the definite integral of the g function. +A similar development at (ξR, ξf) = (1, 1) yields: +E ≈ β−1vL +0 (1, 1)Rw +2 +f Z Ig, +(D4) +which yields vL +0 (ξR, ξf)/vL +0 (1, 1) = ξf/ξR +2 that was used +in the simulations. +[1] S. Li, X. Ding, Z. Mao, Y. Chen, N. Nama, F. Guo, P. 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Am. 61, +1445 (1977). + diff --git a/jtE2T4oBgHgl3EQfIQaF/content/tmp_files/load_file.txt b/jtE2T4oBgHgl3EQfIQaF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ba05adb51af3ab76b60c9cb4242a29de6228376 --- /dev/null +++ b/jtE2T4oBgHgl3EQfIQaF/content/tmp_files/load_file.txt @@ -0,0 +1,697 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf,len=696 +page_content='Numerical simulation of the radiation force from transient acoustic fields: Application to laser-guided acoustic tweezers Shuhan Chen, Qing Wang, Qi Wang, Jia Zhou,∗ and Antoine Riaud† State Key Laboratory of ASIC and System, School of Microelectronics, Fudan University, Shanghai 200433, China (Dated: January 11, 2023) Using pulsed acoustic waves could provide a superior selectivity for microscale acoustic tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' However, the theory for the radiation force of pulsed acoustic waves has only been recently derived and no numerical implementations are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' In this paper, we present a finite-element imple- mentation of this model to simulate the transient acoustic radiation force on small spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' We use the model to simulate laser-guided acoustic tweezers and optimize their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' By enabling numerical simulations of the transient radiation force, this work may accelerate the rational design of pulse-based high-selectivity acoustic tweezers devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' INTRODUCTION The acoustic radiation force (ARF), a steady force created by large-amplitude acoustic waves, is a conve- nient means to achieve micro-object manipulation such as micro-sample separation [1–3] and enrichment [4], cell sorting [5, 6], and single-cell manipulation [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Using transient excitation such as pulses could enable much more precise manipulation than when using time- periodic acoustic fields [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' First, pulsed acoustic ma- nipulation is less disturbed by Rayleigh acoustic stream- ing [8, 9] because radiation force is established much faster than streaming [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Second, using acoustic wave packets allows to localize the acoustic interference pattern and therefore to control the spatial extent of the acoustic trapping region [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Indeed, standing waves exert radiation forces considerably larger than traveling waves (in the small particle limit), which allows to ne- glect the acoustic field outside the interference region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Laser-guided acoustic tweezers (LGAT) [13] use this in- terference principle to create a hybridized radiation force landscape that couples a high-amplitude piezo-generated acoustic field (strong, Z-field) acoustic field and a light- patterned photogenerated acoustic field (weak, L-field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The hybridized field retains the spatial information of the L-field and the strength of the Z-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Despite these potential applications, theoretical and numerical studies of transient acoustic fields are still rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' A crying example of the limited current understanding of transient nonlinear acoustics is the suppression of acous- tic streaming by acoustic pulses [8, 9], where the only model available for transient streaming [14] has been un- able to qualitatively explain experimental observations [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Likewise, there are no numerical schemes to study transient ARF directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' In this paper, we implement the recent generalization of the radiation force theory for small spheres (Gor’kov theory [15]) to transient acoustic fields [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Besides re- quiring that the objects are spherical and much smaller ∗ jia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='zhou@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='cn † antoine riaud@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='cn than the acoustic wavelength at the considered band- width, this theory overlooks micro-streaming and would not be suitable for studying very dense particles in highly viscous fluids (such as copper in glycerol [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' It also requires that scattering events do not overlap and there- fore extreme care must be taken when investigating the response of bubbles or other high-quality acoustic res- onators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Nonetheless, the underpinning assumptions sug- gest that the theory is valid for cells or plastic particles immersed in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The manuscript is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' After summa- rizing the theory of dynamic acoustic radiation force, sec- tion II details a numerical implementation of the dynamic ARF and details the model setup for the simulation of an LGAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Section III briefly presents the acoustic field in an LGAT device and then analyzes the resulting ARF de- pending on the location of the particle and on the phase difference between the laser and the piezoelectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Then, the excitation parameters of the LGAT (pulse duration and laser beam width) are optimized to maximize the ARF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' THEORIES AND METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Transient ARF: Governing equations Our simulations implement the theory of ARF on small rigid spheres in transient acoustic fields of Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Similar to most other theories of radiation pres- sure, the wave peak pressure amplitude pmax is assumed to be small such that pmax/ρ0c2 0 = ϵ ≪ 1 where ϵ is the acoustic Mach number, with ρ0 the quiescent fluid den- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' According to the perturbation method, the fluid quantities can be resolved as p = p0 + p1 + p2, (1a) v = v0 + v1 + v2, (1b) ρ = ρ0 + ρ1 + ρ2, (1c) where subscripts 0 − 2 indicate the perturbation order of each term q0 = q1O(ϵ) = q2O � ϵ2� where q can be any physical field among pressure, density and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Ac- cordingly, time-invariant 0-order terms are interpreted as arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='03678v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='flu-dyn] 9 Jan 2023 2 constant (hydrostatic) contributions, 1st order terms as acoustic contributions, and 2nd order terms as nonlinear effects such as radiation pressure or acoustic streaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Although the theory of radiation force requires knowl- edge of the total acoustic field, Gor’kov theory is conve- niently expressed using the incident acoustic field only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Henceforth, all acoustic quantities will refer to the inci- dent field in the remaining of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Compared to Gor’kov theory, the assumption of single- frequency acoustic fields is extended to wave packets of finite duration τ: all the wave quantities q1 are required to keep the initial state and final state consistent, which means satisfying the condition q1 (0, r) = q1 (τ, r), where r denotes the position vector and the origin of time can be chosen arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Additionally, pulse duration must be short enough to neglect the sphere displacement com- pared to the shortest wavelength, which can be evaluated by 1/τ ≫ fmaxϵ2, where fmax refers to the highest fre- quency in the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' In the case of an inviscid fluid (viscoacoustic and ther- moacoustic boundary layers much thinner than the parti- cle diameter [18, 19]), the acoustic radiation force reads: Frad = −Vp∇U, (2) with the particle volume Vp = 4πR3 p/3 and the general- ized Gor’kov potential: U = ⟨Uinst⟩ = 1 ttot � Uinst dt, (3) where Uinst is interpreted as a fictive (but convenient) instantaneous Gor’kov potential: Uinst = f1V − 3 2f2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (4) Here K = ρ0v1 · v1/2 and V = p2 1/(2ρ0c2 0) represent in- stantaneous kinetic energy and potential energy respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' f1 = 1 − ρ0c0/(ρpcp) is the monopole scattering coefficient and f2 = 2 (ρp + ρ0) /(2ρp + ρ0) is the dipole scattering coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Therefore, computation of the radiation force requires knowing the incident acoustic field, which can be com- puted using transient acoustic pressure models: 1 c2 0 ∂2p1 ∂t2 + ∇2p1 = 0, (5) from which the fluid acceleration can be deduced using the Euler equation: a1 = − 1 ρ0 ∇p1, (6) and the velocity: v1 = � a1 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (7) Under these hypotheses, the general Gor’kov approach can be used for calculating ARF in such transient situa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Transient ARF: Numerical implementation The numerical model for the simulation of hybridized acoustic fields was configured in the commercial finite- element-method software COMSOL Multiphysics® ver- sion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='4, and the parameters used are given in Ta- ble I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The model uses the “Pressure Acoustics, Transient” module to solve the first-order acoustic pressure field in the fluid domain (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (1b) and to deduce the accelera- tion and velocity fields with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The settings for transient wave solving follow the approach suggested by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Transient acoustic field studies are necessarily associ- ated with bandwidth issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' On the one hand, the exci- tation signal (such as a short laser pulse) may have a very wide bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' On the other hand, integration of the acoustic partial differential equation is often done with a constant time step and the grid imposes constraints on the minimum wavelength that can be resolved (that is, the maximum frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The Courant-Friedrichs-Lewy (CFL) condition provides guidelines to set the time step ∆t depending on the maximum mesh size h [14, 20]: CFL = c∆t h , (8) where c is the wave speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Using the default second- order (quadratic) mesh elements, the recommended CFL number is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Thus, given a maximum frequency fmax that needs to be resolved, the CFL yields the time step ∆t = CFL/(Nfmax) where N is the number of elements per local wavelength λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Due to the period doubling of the Gor’kov potential (terms in p2 1 and v2 1), we choose N = 12 instead of the more common N = 6 in linear acoustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Boundary conditions must be designed carefully ac- cording to this bandwidth constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The most straight- forward is to directly set an acceleration that fits in the simulated frequency bandwidth as the boundary condi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' However, in some cases, the velocity needs to be specified instead (for instance when the velocity is known from laser Doppler vibrometer measurements) which can yield to step-wise acceleration if there is a slope-break in the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' To address this problem, the input veloc- ity is first multiplied by a smoothing window, and then derived to obtain the acceleration boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The velocity field is computed (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (7)) by coupling the pressure acoustic model to a “Distributed Ordinary Dif- ferential Equation (DODE)” module in COMSOL with the damping factor set to 1 in DODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The first-order velocity is then used to calculate the instantaneous Gor’kov potential Uinst via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (4) along with the first-order acoustic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Similarly, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (3) is implemented in the “Distributed Ordinary Differen- tial Equation (DODE)” module to obtain U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Finally, the ARF is obtained as the gradient of U (using the COMSOL function diff()).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Given the small size of the geometry, computer memory is not a constraint in this study, and all the fields are computed using a single fully- 3 coupled time-dependent solver for simplicity, but the per- turbation approach of the model could allow using a seg- regated solver that solves the fields one after another for each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Summary of simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Electroacoustic wave parameters Frequency f Z 10 MHz Velocity magnitude vZ 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='83 m/s Number of shots nZ 5 Density ρz 4640 kg/m3 Laser-induced acoustic wave parameters Frequency f L 10 MHz Velocity magnitude vL 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='6 m/s Radius of spot waist Rw 20 µm Fluid Density ρ0 1050 kg/m3 Speed of sound c0 1500 m/s Domain width (radius) Rd 375 µm Domain height hd 200 µm Particle Density ρp 1050 kg/m3 Radius Rp 10 µm Mesh & Solver CFL number CFL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='5 Units per local wavelength N 12 Others Simulation time tsim 2 µs Duration between pulse series ttot 20 µs C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Simulation of an LGAT: Initial and boundary conditions In an LGAT (schematic shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 1), two acoustic sources interfere to generate an acoustic trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' A piezo- electric acoustic source (strong field, labeled with the superscript “Z”) yields a plane traveling wave of high amplitude which creates no in-plane acoustic radiation force due to symmetry constraints (the wave is in-plane invariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' This wave interferes with a weaker acoustic wave (weak field, labeled with the superscript “L”) gen- erated by the photoacoustic conversion of a laser pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The combined incident pressure and velocity fields read p1 = pL 1 + pZ 1 and v = vL 1 + vZ 1 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Accord- ing to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (3) and (4), this yields a Gor’kov potential with cross terms U = U ZZ + U ZL + U LL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The vanishing gradient of spatially invariant U ZZ leads to a negligible in-plane force, while U LL is negligible due to the small photoacoustic conversion efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' As a result, only the hybridized potential U ZL dominates, and the force is well approximated by: Frad = −Vp∇U ≈ −Vp∇U ZL = −Vp∇ � U ZL inst � , (9) with U ZL inst = f1pZ 1pL 1/(ρ0c2 0)− 3 2f2ρ0vZ 1 ·vL 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' We note that Uinst is twice larger than in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (4) due to the binomial coefficients in the expansion of p2 1 and v1 · v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Laser-guided acoustic tweezers (LGAT) structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The particle is manipulated by the combination of a weak and a strong acoustic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The interference between the two fields yields a radiation force with spatial characteristics similar to the weak field but with a much higher amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The L-field is generated by the photoacoustic conversion of a pulsed laser beam on the gold nanoparticle-polydimethylsiloxane (AuNP- PDMS) composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The composite selectively absorbs green light and is transparent to other wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The Z-field is generated by piezoelectric conversion of an electric signal (alternating current, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=') in the LiNbO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The electrodes (indium tin oxide, ITO) are transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' As a result, the whole device is transparent to most optical wavelengths ex- cept green, and the particles under manipulation can be di- rectly visualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' While we are mainly focused on the acoustic radiation pressure, we note that acoustic streaming plays a negligi- ble role in this study due to the particle radius of 25 µm used in the experiments [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' For fluids in the channel, thermophoresis and thermal agitation can also be factors driving the particles [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' To rule out this possibility, we have turned off the bulk acoustic wave (BAW) from Z- source and observed no motion of particles despite the laser heating [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Therefore, thermal effects are also ne- glected in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The other parameter values are given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Most of these parameters are selected based on the typical experimental values while they are sometimes simplified to facilitate dimensionless analysis (for instance f Z is rounded to 10 MHz instead of the exact resonance frequency of the crystal used in experiments (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='6 MHz)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Further, an important goal of the paper is to estimate what would be the upper limit of the radi- ation force of the LGAT if suitable laser and electronic excitation were provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' For the electroacoustic field, a simple estimate is obtained by using the definition of the piezoelectric coupling coefficient K2: 1 2C(Ed)2K2 ≃ 1 2ρzAdvZ 0 2, (10) where d is the thickness and A is the area of the dielectric crystal, together with the capacitance C = (εrε0A)/d and the field strength E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' On this basis, an estimation of the velocity is derived as vZ 0 = KE � εrε0/ρz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' This velocity is independent of the excitation frequency and is limited by the coercivity field strength of the crystal, which is of the order of 21 MV/m for LiNbO3 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The Y-36° cut has a K2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' For the sake of simplicity, the material anisotropy is overlooked and we consider a Chamber particle water AuNP-PDMS PDMS ITO a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='C LiNbO ITO laser4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Model setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (a) Sketch of the two-dimensional axial symmetry computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The purple domain is used for the simulation of the Z-field and the calculation of the radiation force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The gray domain is used for the calcula- tion of the L-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (b) Input signals applied to the bottom boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Input signals are depicted by velocities varying with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Grey and green curves represent Z-signal (elec- troacoustic BAW) and L-signal (photoacoustic pulse) respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The zoom-in shows the signal duration overlap under typical relative phases (∆ϕ = ±π/2) within the time window of a period (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='47 µs to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='62 µs ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' relative permittivity of the order of εr ≈ 40, which yields a maximum velocity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='83 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' To satisfy U LL ≪ U ZL as required by the hybridized- fields theory, the maximum excitation velocity of the pho- toacoustic field is set to 60 cm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' We note that the ma- terial damage threshold could in principle allow higher velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' More details about the criterion are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The laser spot being axisymmetric and the Z-wave be- ing spatially-invariant, we use a two-dimensional axisym- metric model (cylindrical coordinates system) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' We restrict our analysis to a rectangular cross-section (purple part in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 2(a)) of width Rd = nλλZ/2 = 375 µm (with nλ = 5) and height hd = 200 µm in the vertical r-z plane for a 10 MHz wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The width is chosen large enough to ensure that the tweezers yield a negligible force at that distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' In experiments, the particles are contained in a poly- dimethylsiloxane (PDMS) microchannel filled with wa- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' PDMS and water have a similar acoustic impedance (plane wave reflection coefficient R = 18%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' In order to reduce simulation time, the simulation domain size is minimized by neglecting acoustic reflections on the PDMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Instead, plane wave radiation boundary condi- tions are used for the Z-wave, and spherical wave radi- ation ones are used for the L-wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' A virtual boundary of radius Rvb = � R2 d + h2 d + λZ/2 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 2(a) is constructed to facilitate the setting of spherical wave radiation boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' More refined boundary conditions accounting for reflections in the PDMS-water surface and shear waves in the PDMS are discussed by Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The transducer is not explicitly modeled, instead, the piezoelectric Z-excitation is implemented as an r- invariant acceleration: aZ = aZ 0sin � ωZt − ϕZ� Λ �ωZt − ϕZ 2π � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (11) The triangular window Λ(·) with the trigger range of [0, nZ] simulates the resonance onset of the piezoelec- tric with a train of nZ = 5 cycles, (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' We note that longer excitation could be possible but would be more sensitive to reflections at the lateral edge of the wafer that would induce standing waves (and violate the in-plane invariance of the Z-wave).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The relationship ωL = ωZ exists unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Likewise, the photoacoustic excitation is described by an acceleration with a Gaussian intensity profile, since the photoacoustic vibration is well approximated by a Gaussian profile according to our experimental measure- ments [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' We set the radius of the beam waist Rw close to the experimental value (20 µm), producing a normal acceleration given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (12), where the amplitude reads aL 0 = vL 0 /ωL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' A rectangular window function modu- lates the sinusoidal signal to simulate the experimentally- detected pulsed L-signal (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 2(b)): aL = ∂vL 0 ∂t = aL 0e−(r/Rw)2 ∂ ∂t � sin � ωLt − ϕL� Π �ωLt − ϕL 2π �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (12) (a) BC: spherical wave radiation BC: plane wave radiation 2/2 BC: symmetry R fluid:domain Rd BC: normal acceleration (b) ×108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='8 Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='2 (m/ 0 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='8 0 2 t (us) 元5 Here the rectangular window Π(·) is set to 1 over the interval [0, 1/2] to simulate a pulse of duration τ L = π/ωL (here τ L = 50 ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' A smoothing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='4 is introduced to introduce frequen- cies above fmax = f Z that would trigger numerical insta- bilities (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The smoothing process intro- duces small overshoots at the beginning and the end of the laser waveform function with an amplitude negligible compared to the main pulse peak (≈ 20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Both excita- tions have an adjustable phase ϕ that can be set in exper- iments using an adjustable delay to a common reference trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' By default, ϕZ = 0 and ϕL = (2nZ + 1/2)π such that the two excitation waveforms are in-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' When the phase between Z and L is adjusted, ϕZ is the adjust- ment variable (ϕL remains unchanged) and an empirical reference ϕ0(r) is introduced such that ∆ϕ = ϕZ − ϕ0 vanishes when the force is 0 at the measurement point r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' This will be discussed in more detail in section III B 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' RESULTS In this section, we illustrate the working of the laser- guided acoustic tweezers and discuss potential directions for parameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Acoustic field As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 2(b), the simulation starts at t = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='2tsim = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='40 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' At t = 0, the piezoelectric begins to emit a BAW (Z) wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The photoacoustic pulse (L- signal) hits the substrate at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='52 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Therefore, the observation of time evolution begins at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='52 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Acoustic pressure of the L-field on the central axis of the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 2, the laser beam hits the substrate at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='52 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The Z-field is a plane traveling wave and remains sim- ilar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 2(b) as it travels in the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The L- wave generated by photoacoustic conversion is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' It is essentially a spherical wave that behaves sim- ilarly in the z- and r-directions (r-profile available in Ap- pendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The L-acoustic pressure reaches its maximum amplitude immediately after generation at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='54 µs and decays by 80% by the time it exists in the simula- tion domain (t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='66 µs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Acoustic radiation pressure In manipulation experiments, the particle is confined in the manipulation chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Therefore, radial forces tend to be more important than axial ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Hence, un- less specified otherwise, the ARF will refer to F ZL r = −Vp ∂U ZL ∂r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Build-up of the Gor’kov potential Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 4 shows the build-up of the acoustic radiation pres- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The pressure of the Z and L-waves are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 4(a): the Z-wave is a traveling plane wave, and the L-wave is similar to a spherical wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Despite being a virtual quantity, the instantaneous po- tential Uinst to visualize the hybridization of the L and Z fields during the acoustic radiation pressure build-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' For example, the region where Uinst < 0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 4(b) cor- responds to the mixing of pL 1 and pZ 1 of opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' This instantaneous Gor’kov potential grows over time, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' As the spherical wavefront (L) propagates, the inhomogeneous Gor’kov potential region spreads to the entire simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' A cross-section of the hybridized Gor’kov potential U ZL experienced by a particle resting on the bottom wall (z = Rp) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The potential is mainly concentrated at the location of the laser spot, which ensures good selectivity for acoustic trapping and manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' We note that our model neglects the effect of walls, which needs to be accounted for using a more advanced theory (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' [26] for the monofrequency case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Effect of the particle location The effect of the particle location is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 5, with ∆ϕ = π/2 selected to maximize the radiation force (this will be interpreted in section III B 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The closer the observation position z is taken to the lower edge, the larger the ARF peak generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The concentration of the pulse energy is also reflected in the ARF curves shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 5(a) and (b), where the ARF peaks show a decay- ing trend when moving away from the excitation source from either the r-direction or the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 5, not only the force becomes very weak when the particle is located near the top of the channel, but the selectivity (ratio of primary force maximum to the sec- ondary one) also deteriorates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' It is therefore important in LGAT design to ensure that the particles are relatively ×105 (us) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='66 a 4 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='64 Acoustic Pressure 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='62 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='6 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='58 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='56 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='54 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='52 0 50 100 150 200 z-coordinate (um)6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Time-dependent field evolution with ∆ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Pictures are taken at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='54, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='62, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='70, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='78 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (a) Super-imposed acoustic pressure field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The blue-red color represents the L-field pressure and the contour represents the Z-field pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (b) Instantaneous Gor’kov potential Uinst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (c) Time-averaged Gor’kov potential U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (d) Cross-section of the magnitude of U at z = Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Variation of the hybridized acoustic radiation force F ZL r at various heights in the manipulation chamber when ∆ϕ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' close to the laser spot, which may limit the biocompati- bility of the manipulation, unless holographic techniques are used [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' In the following, the force is maximized by assuming that the particle rests at the bottom of the channel (z = Rp) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Effect of the phase difference between Z and L fields A distinctive feature of LGAT is the possibility to com- mute between attractive and repelling forces by adjust- ing the electronic delay between the triggers of the laser source and the electric signal generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' In the simula- tion, this is implemented by changing the relative phase difference ∆ϕ between the Z and L fields, as shown for a particle located at z = Rp = 10 µm, r = 3Rp = 30 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Following a convention proposed in our previ- ous paper, we add an offset ϕ0(r) to ∆ϕ such that the ARF cancels for ∆ϕ = 0 [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' A better alternative will be proposed later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (a) t =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='54μs t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='70 μs t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='78 μs ApL Apz t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='62 μs Pa Pa Z-coordinate ( ×105 ×106 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='24 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='64 0 100 200 300 r-coordinate (um) (b) 200 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='54 μs t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='62μs t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='78μs Uinst t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='70 μs J/m3 z-coordinate 200 100 1-200 0 100 200 300 r-coordinate (um) (c) (um) 200 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='54 μs t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='62 μs t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='70 μs t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='78 μs U J/m3 z-coordinate ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='2 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='2 0 100 200 300 r-coordinate (um) (d) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='54 μs t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='62μs t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='70 μs t =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='78 us J/m3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='1×10-12 0 5 10 15 (N) 20 25 30 35 2R, 40 R 45 R,/2 50 0 100 200 300 r-coordinate (um)7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Hybridized acoustic radiation force F ZL r at z = Rp = 10 µm, r = 3Rp = 30 µm depending the phase difference ∆ϕ between the Z and L fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The variation of the ARF at fixed z = Rp = 10 µm but with a varying phase is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Here, the phase offset is kept at 0 such that ∆ϕ = ϕZ every- where.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Although at r = 3Rp = 30 µm the force behaves consistently with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 6 (strongest pull at ∆ϕ = π/2, strongest push at ∆ϕ = −π/2 and vanishingly small at ∆ϕ = 0, it can be seen that points at half a wavelength away (such as r = 105 µm) behave in the opposite fash- ion and points at a quarter wavelength distance (such as r = 62 µm) behave in phase quadrature (strongest pull at ∆ϕ = π, strongest push at ∆ϕ = 0 and vanishingly small at ∆ϕ = ±π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' This emphasizes that the exact delay between L and Z fields depends not only on the instruments trigger but also on the acoustic propagation of both fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The latter is relative to the exact point in space where the ARF is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' This issue is ad- dressed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' We also note that the force is not exactly periodic over time due to the slow variations of the wave envelope (the wave amplitude is not the same for ∆ϕ = −π and ∆ϕ = π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' A phase-independent factorization of the hybridized Gor’kov potential In the previous section, we have shown that the phase difference ∆ϕ is an important factor controlling the ARF of LGAT, but that the relationship between the force di- rection and ∆ϕ depends on the spatial location of the particle being manipulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Hence, gaining a perfect knowledge of the ARF of an LGAT would require re- peating ARF calculations for all phases ∆ϕ ∈ [−π, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Here, we show how to reduce this large number of cal- culations to only two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' [13] gives a way to obtain the hybridized potential U ZL by setting up an analyti- cal Z-field wave with constant amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Then U ZL is expressed as a linear combination of two basis potentials FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Variation of the hybridized acoustic radiation force F ZL r at z = Rp depending on the phase difference ∆ϕ be- tween the Z and L fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The line color indicates the phase while the line width grows with the phase in order to visually distinguish between ∆ϕ = −π and ∆ϕ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' weighted by trigonometric functions of ∆ϕ: ˆU ZL = Φccos ∆ϕ + Φssin ∆ϕ, (13) with the basis potentials Φc (r) = � pZ 1|∆ϕ=0EL� and Φs (r) = � pZ 1|∆ϕ=π/2EL� independent of ∆ϕ, where L- component EL = f1pL 1/(ρ0c2 0) − 3 2f2vL 1z/c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Here only the z-direction component of vL 1 is considered due to z-propagating-only vZ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' ˆU ZL obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (13) is strictly valid only for monofrequency Z-wave but is as- sumed here to be a good approximation of the actual potential U ZL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' In the simulations, Φc (Φs) is obtained by setting ∆ϕ = 0 (∆ϕ = π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The approximate poten- tial can then be computed from these two basis potentials according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 8(a) shows the basis potentials at z = Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Their values can be recorded in the entire simulation do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The quality of the approximation is evaluated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 8(b): the approximated potential matches U ZL (from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (3)) well, although the relative error for ∆ϕ = −π increases by ∼ 8% compared to the one of ∆ϕ = π/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' This is likely because the Z-wave has a triangular enve- lope of varying amplitude which is not considered in the theory of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Optimization of operating parameters The main weakness of current LGAT implementations is that they yield a very small force that results in dis- placement speeds of a few micrometers per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' In the following, we optimize two parameters readily adjustable in experiments: (1) the ratio of the L-wave frequency (reflecting the laser pulse duration) with respect to the Z-wave frequency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=', ξf = f L/f Z (which can be ad- justed on most pulsed laser sources);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (2) the size of the ×10-12 r = 30 um, z = 10 um 25 20 15 10 5 (N) 0 5 Z 10 F 15 20 25 30 35 1 0 1 △0 (元)△ (元) 30 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='8 20 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='6 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='4 505° (N) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='2 0 05252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='6 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='8 35 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='1 40 0 100 200 300 r-coordinate (um)8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Approximate Gor’kov potential ˆU ZL with ∆ϕ decou- pled (a) Constant basis potentials Φc and Φs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (b) Comparison between ˆU ZL and U ZL for ∆ϕ = −π and π/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' pulse source, expressed as the radius of the laser spot Rb (which can be adjusted by changing the microscope objective magnification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' We assume that ξf is adjusted by changing f L with f Z fixed at 10 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' In this section, the conditions ∆ϕ = π/2 and z = Rp are retained, and the radiation force magnitude Fr max is discussed based on the maximum absolute values of the ARF in the r- direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' To reflect experimental constraints, the comparisons are done at constant laser peak energy (that would be equivalent to adjusting the laser energy to avoid material damage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Photoacoustic generation theory shows that the photoacoustic wave amplitude is proportional to the laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Hence, we adjust the photoacoustic wave amplitude as vL 0 ξf (see Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' In addition, the laser phase ϕL is adjusted to keep the peak aligned with the electroacoustic field peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 9 illustrates the effect on ARF of modulating ξf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The ARF increases fast as the pulse shortens until it reaches a saturation value of Fr max ∼ 44 pN when the pulse frequency satisfies (f L ∼ f Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Afterward, the in- crease flattens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' While the marginal improvement past f L ∼ f Z might seem attractive for improved versions of the LGAT, we note that the short wavelength at high frequencies violates Gor’kov’s assumption of an object size much smaller than the wavelength, and the onset of resonance can yield to an uncontrollable force direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Hybridized acoustic radiation force F ZL r for various pulse duration (f L/f Z) while keeping the laser pulse energy constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Next, we consider varying the laser spot size, for instance by using objectives of varying magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Here, Rb is introduced to distinguish the adjusted spot radius from the original radius Rw used in the other parts of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The ratio ξR = Rb/Rw is defined for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' To maintain constant pulse energy, the scaling vL 0 /ξ2 R neglecting the loss of transmission of high- magnification objectives is used (detailed derivation in Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 10(a) and (b) show the evolution of the ARF de- pending on the pulse source radius when ∆ϕ = 0 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 10(c) and (d) shows the evolution of the ARF de- pending on the pulse source radius when ∆ϕ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' It is found that ∆ϕ = π/2 yields the largest force regardless of the frequency and beam radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Optimizing the latter two parameters (f L = 14 MHz (pulse duration of 31 ns), Rb = 10 µm), we expect a force of ∼ 80 pN, well within the range of other acoustic tweezers, but that would have the extra advantage of being controlled by a light pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' PERSPECTIVE In the model, we have only discussed the case of small particles in an ideal fluid domain, and the properties of the wave sources are relatively simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Therefore, the model deserves further extensions, such as the introduc- tion of thermoviscous effects and interaction with walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Furthermore, we have assumed in the model that the radius of the particle was small enough to be consis- tent with the Gor’kov approximation [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' However, the LGAT experiments [13] of manipulation of ∼ 25 µm ra- dius particles have unveiled a much larger force, presum- ably due to the onset of resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Therefore, a theory for transient ARF valid for arbitrary particle radii would be highly desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' a (un) 200 Φc (Ag = -元/2) z-coordinate 100 0 100 200 300 r-coordinate (um) Φs (A = -π) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='06 Potential (J/m3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='04 (=-元) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='06 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='0(A = -元) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='08 U(Aβ = 元/10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='0(AP = 元/10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='14 0 100 200 300 r-coordinate (um)×10-12 40 35 30 (N) 25 20 15 10 5 0++ 0 109 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Hybridized acoustic radiation force F ZL r for various pulse durations (expressed as frequencies of 6, 8, 10, 12, and 14 MHz) and beam radii, while keeping the beam energy constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (a) and (b) presents the force versus the beam radius Rb and the dimensionless radius Rb/λZ for ∆ϕ = 0 while (c) and (d) presents those for ∆ϕ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' CONCLUSION Despite the availability of a theoretical expression for the ARF of short pulses, no finite element models had been implemented so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' In this paper, we have shown that such a simulation could be carried out by (i) using time-explicit acoustic pressure simulations to compute the acoustic field, (ii) defining a virtual instantaneous Gor’kov potential, and (iii) integrating it over time to ob- tain the time-averaged Gor’kov potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' We have then used our code to simulate laser-guided acoustic tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Our simulations show that the behavior of the LGAT can be captured by two conjugated Gor’kov potentials Φc and Φs weighted by trigonometric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Optimization of operating parameters shows that the force increases when reducing the laser spot size, while the gain from shorten- ing the pulse duration is limited (∼ 10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Finally, based on the dielectric breakdown limit of the piezoelectric, we estimate the maximum LGAT force on 10 µm polystyrene particles in water to ∼ 40 pN, which is comparable to other acoustic tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Beyond LGAT, the availability of a numerical model may accelerate the development of time-dynamic acoustic tweezers with superior dexterity compared to their monochromatic counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' a ×10-12 ×10-12 6 MHz 6 MHz 24 24 0- 8 MHz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='-0- 8 MHz 22 22 --→- 10 MHz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='→- 10 MHz 20 20 -- 12 MHz +- 12 MHz 18 18 +: 14 MHz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='-+- 14 MHz 16 16 N 14 14 12 12 F 10 10 8 8 6 6 4 4 2 2 0 0 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='5 Rb (um) Rb/aZ (d) C ×10-12 ×10-12 6 MHz 6 MHz 56 0- 8 MHz --0- 8 MHz →- 10 MHz --→- 10 MHz 385844 12 MHz -- 12 MHz +- 14 MHz +- 14 MHz Z 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='5 Rb (um)10 ACKNOWLEDGMENTS This work was supported by the National Natural Sci- ence Foundation of China with Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 12004078, 61874033, and 62274039;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' the State Key Lab of ASIC and System, Fudan University with Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 2021MS001, 2021MS002, and 2020KF006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' and the Science and Technology Commission of Shanghai Municipality with Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 22QA1400900 and 22WZ2502200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Appendix A: Magnitude comparison between U LL and U ZL The simulation method follows the hypothesis of the hybridized-fields theory [13], which expects a relatively low value of U LL compared to U ZL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' U LL is independent of ∆ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Here we evaluate the effect of U LL with the max- imum U ZL induced by adjusting ∆ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 11, the maximum U LL reaches less than 10% of the maximum U ZL, which validates our as- sumption for the current simulation settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Magnitude comparison between U LL and U ZL with ∆ϕ = ±π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Appendix B: Effects of input signal smoothing Since the acceleration is derived from the derivation of the velocity, we will get an input aL with a sharp variation at the beginning and the end of the signal (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 12(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Large changes tend to generate numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' To minimize the issue, we introduce a transition region of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='4ωL/2π on each side of the rectangular window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 12(b), we check that the deviation of the ARF resulting from the smoothing is moderate when compared to the non-smoothed function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Comparison between the sharp signal and the smoothed signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (a) Signal input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (b) F ZL r on z = Rp with ∆ϕ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Appendix C: Isotropy of the laser pulse The consistency of the variation trend and magnitude of the acoustic field in the r-direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 13) with that in the z-direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 3) reflects the spherical wave ap- proximation condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' However, in the near-field zone, the acoustic pressure change in the r-direction is sharper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Such difference in propagation characteristics of different directions originates from the normal direction of the ini- tial acceleration, while the rapid attenuation of acoustic fields in both directions reflects the strong concentration of acoustic energy obtained by photoacoustic generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Appendix D: Scaling velocity to maintain a constant pulse energy When changing operating parameters in the simula- tions, the laser pulse energy is assumed to remain con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' This is achieved by scaling the L-velocity accord- ing to the Vashy-Buckingham Pi-theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The velocity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='3 —UZL(=-/2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='2 —UZL(△=元/2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='15 ULL 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='35 0 100 200 300 r-coordinate (um)(a) ×107 4 Smoothed *** Original 3 2 (m/s2) 1 0 XXXXXXXXXXX 上 1 a 2 3 0 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='6 t (us) (b) ×10-12 5 0 5 10 (N) 15 20 25 30 Smoothed 35 Original 40 0 100 200 300 r-coordinate (um)11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Propagating properties of L-acoustic pressure vary- ing with r on the lower edge (z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The color bar from light yellow to dark red represents the change of time from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='52 µs to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content='66 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' is assumed to read: vL(f Lt, r/Rb) = vL 0 (ξR, ξf)g(f Lt, r/Rb), (D1) where g is an arbitrary integrable function that can be obtained in our simulations by integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The goal of the following calculation is to determine vL 0 (ξR, ξf)/vL 0 (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' The pulse energy from the laser reads E = � Atot � ttot ε dS dt, where ε is the pulse energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Substituting ε = β−1vL, with β the photoacoustic con- version coefficient (assumed to be constant for the lim- ited frequency and power range studied here) and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (D1), we get: E ≈ β−1vL 0 (ξR, ξf) � ∞ � ∞ g(f Lt, r/Rb) dS dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' (D2) The right-hand side integral is obtained by assuming that the simulation domain is large enough to completely en- compass the laser spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Changing the integration vari- ables to f Lt and r/Rb, we get: E ≈ β−1vL 0 (ξR, ξf)Rb 2 f L Ig, (D3) where Ig is the definite integral of the g function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' A similar development at (ξR, ξf) = (1, 1) yields: E ≈ β−1vL 0 (1, 1)Rw 2 f Z Ig, (D4) which yields vL 0 (ξR, ξf)/vL 0 (1, 1) = ξf/ξR 2 that was used in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Ding, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Mao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Chen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE2T4oBgHgl3EQfIQaF/content/2301.03678v1.pdf'} +page_content=' Nama, F.' metadata={'source': 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b/kb_25/content/tmp_files/dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%2520metagenomics%2520roadmap%2520to%2520the%2520uncultured%2520genome%2520diversity%2520in%2520hypersaline%2520soda%2520lake%2520sediments.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..78aaa183d25fe30879e2b6e9df02ed8ea45f0520 --- /dev/null +++ b/kb_25/content/tmp_files/dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%2520metagenomics%2520roadmap%2520to%2520the%2520uncultured%2520genome%2520diversity%2520in%2520hypersaline%2520soda%2520lake%2520sediments.pdf.txt @@ -0,0 +1,1769 @@ +RESEARCH +Open Access +A metagenomics roadmap to the +uncultured genome diversity in hypersaline +soda lake sediments +Charlotte D. Vavourakis1 +, Adrian-Stefan Andrei2†, Maliheh Mehrshad2†, Rohit Ghai2, Dimitry Y. Sorokin3,4 +and Gerard Muyzer1* +Abstract +Background: Hypersaline soda lakes are characterized by extreme high soluble carbonate alkalinity. Despite the +high pH and salt content, highly diverse microbial communities are known to be present in soda lake brines but +the microbiome of soda lake sediments received much less attention of microbiologists. Here, we performed metagenomic +sequencing on soda lake sediments to give the first extensive overview of the taxonomic diversity found in these complex, +extreme environments and to gain novel physiological insights into the most abundant, uncultured prokaryote lineages. +Results: We sequenced five metagenomes obtained from four surface sediments of Siberian soda lakes with a pH 10 and +a salt content between 70 and 400 g L−1. The recovered 16S rRNA gene sequences were mostly from Bacteria, even in +the salt-saturated lakes. Most OTUs were assigned to uncultured families. We reconstructed 871 metagenome-assembled +genomes (MAGs) spanning more than 45 phyla and discovered the first extremophilic members of the Candidate Phyla +Radiation (CPR). Five new species of CPR were among the most dominant community members. Novel dominant +lineages were found within previously well-characterized functional groups involved in carbon, sulfur, and nitrogen +cycling. Moreover, key enzymes of the Wood-Ljungdahl pathway were encoded within at least four bacterial phyla +never previously associated with this ancient anaerobic pathway for carbon fixation and dissimilation, including the +Actinobacteria. +Conclusions: Our first sequencing effort of hypersaline soda lake sediment metagenomes led to two important +advances. First, we showed the existence and obtained the first genomes of haloalkaliphilic members of the CPR +and several hundred other novel prokaryote lineages. The soda lake CPR is a functionally diverse group, but the most +abundant organisms in this study are likely fermenters with a possible role in primary carbon degradation. Second, +we found evidence for the presence of the Wood-Ljungdahl pathway in many more taxonomic groups than those +encompassing known homo-acetogens, sulfate-reducers, and methanogens. Since only few environmental metagenomics +studies have targeted sediment microbial communities and never to this extent, we expect that our findings are relevant +not only for the understanding of haloalkaline environments but can also be used to set targets for future studies on marine +and freshwater sediments. +Keywords: Soda lake sediments, Metagenomics, Haloalkaliphilic extremophiles, Candidate Phyla Radiation, Wood-Ljungdahl +pathway +* Correspondence: G.Muijzer@uva.nl +†Adrian-Stefan Andrei and Maliheh Mehrshad contributed equally to this +work. +1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, +Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, +University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the +Netherlands +Full list of author information is available at the end of the article +© The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 +International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and +reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to +the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver +(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. +Vavourakis et al. Microbiome (2018) 6:168 +https://doi.org/10.1186/s40168-018-0548-7 + +MicrobiomeBackground +Soda lakes are evaporative, athallasic salt lakes with low cal- +cium and magnesium concentrations and a high-alkaline +pH up to 11 buffered by dissolved (bi-) carbonate ions [1]. +They are constrained to arid regions across the globe, +mainly the tropical East African Rift Valley [2], the Libyan +Desert [3], the deserts in California and Nevada [4], and the +dry steppe belt of Central Asia that spans to southern Si- +beria, north-eastern Mongolia, and Inner Mongolia in +China [1]. On top of the extreme salinity and alkaline pH, +the Eurasian soda lakes experience extreme seasonal +temperature differences, causing highly unstable water re- +gimes and fluctuating salinities [5]. Yet, soda lakes harbor +diverse communities of haloalkaliphilic microbes, mostly +prokaryotes that are well adapted to survive and grow in +these extreme environments and consist of similar func- +tional groups in soda lakes around the world [1, 2, 6]. The +relative abundance of different groups is typically governed +by the salinity of the brine [1, 7, 8], and microbial-mediated +nutrient +cycles +become +partially +hampered +only +at +salt-saturating conditions [1]. +So far, all characterized prokaryotic lineages cultured +from soda lakes comprise over 70 different species within +more than 30 genera [1, 6, 9, 10]. From these, only a lim- +ited number of genomes have been sequenced today, +mostly from chemolithoautotrophic sulfur-oxidizing bac- +teria belonging to the genus Thioalkalivibrio (class Gam- +maproteobacteria) [1, 11, 12]. It is well established that +metagenomics enables the recovery of genomes and the +identification of novel genetic diversity where culturing ef- +forts fail [13, 14]. In recent years, next-generation sequen- +cing has recovered a massive number of genomes from +previously unknown groups of prokaryotes [15, 16], +including a strikingly large and diverse group called +“Candidate Phyla Radiation” (CPR), only distantly related +to other cultured bacterial lineages [17]. Previously, we +conducted a metagenomics study on soda lakes and re- +constructed novel genomes from uncultured Bacteroidetes +and “Candidatus Nanohaloarchaeaota” living in hypersa- +line Siberian soda brines [7]. Here, we turned our atten- +tion to the far more complex prokaryotic communities +living in the sediments of the hypersaline soda lakes from +the same region. We give a broad overview of all the +taxonomic groups sequenced and focus on the metabolic +diversity found in the reconstructed genomes of the most +abundant, uncultured organisms. +Results +Overall prokaryote community structure +The salinities from the studied soda lakes ranged from +moderately hypersaline (between 70 and 110 g L−1) to +salt-saturated (400 g L−1 salt). The soluble carbonate al- +kalinity was in the molar range, and the pH in all lakes +was around ten (see Additional file 1: Table S1). To give +an overview of the overall prokaryotic community com- +position in each of the samples, we looked at the taxo- +nomic classification of 16S rRNA genes recovered both +by amplicon sequencing and direct metagenomics se- +quencing (Fig. 1, see also Additional file 2: Figure S1; +Additional file 3). The prokaryotic communities of all +five sediment samples were highly diverse and consisted +mostly of uncultured taxonomic groups. Bacteria were +more abundant than Archaea, regardless of the salinity +of the overlaying brine [7] (Fig. 1). Euryarchaeota were +the second and third largest group in the sediments of +the two salt-saturated lakes comprising ~ 10 and ~ 20% +of the 16S rRNA genes in the metagenomes. Most +Euryarchaeota-related OTUs detected by amplicon se- +quencing belonged either to the uncultured Thermoplas- +mata group KTK 4A (SILVA classification) or the genera +Halohasta and Halorubrum (class Halobacteria). In ac- +cordance with cultivation-dependent studies [6], most +OTUs assigned to methanogens were from the class +Methanomicrobia, +especially +the +lithotrophic +genus +Methanocalculus (up to ~ 3%) and the methylotrophic +genus Methanosalsum (Additional file 3). +The varying ratio of the three dominant bacterial groups, +Firmicutes, Bacteroidetes (including the newly proposed +phyla +Rhodothermaeota +and +Balneolaeota +[18]), +and +Gammaproteobacteria, showed no clear trend in relation to +the salinity in the lakes, but when Firmicutes were domin- +ant, Bacteroidetes were less abundant and vice versa. Most +Firmicutes belonged to the order Clostridales. Uncultured +members from the family Syntrophomonadaceae had a +relative abundance of more than 5% in all five metagen- +omes and comprised in two lakes even ~ 11–20% of the +recovered amplicon sequences. The second most abundant +Firmicutes order was Halanaerobiales, particularly the +genus Halanaerobium (family Halanaerobiaceae) and un- +cultured members of the Halobacteroidaceae. The majority +of Bacteroidetes-related OTUs could not be assigned down +to the genus level. The uncultured ML635J-40 aquatic +group (order Bacteroidales) comprised at least 5% of all five +prokaryotic communities. This group has been previously +found to be abundant in Mono Lake [4] (a soda lake) and +in an anoxic bioreactor degrading cyanobacterial biomass +under haloalkaline conditions [19]. Two other highly abun- +dant (up to ~ 8%) uncultured groups from the class Balneo- +lia (proposed new phylum Balneolaeota [18]) were also +detected in other soda lakes before [3, 4]. Within the Gam- +maproteobacteria, the genus Thioalkalivibrio was abundant +(~ 3% of the total community), but also uncultured +members of HOC36 were prevailing at moderate salinities. +Members of the Deltaproteobacteria, Alphaproteobacteria, +and Chloroflexi comprised up to ~ 10% of the detected 16S +rRNA gene in some of the metagenomes. The GIF9 family +of the class Dehalococcoidia was among the top three most +abundant OTUs in two lakes. The extremely salt-tolerant +Vavourakis et al. Microbiome (2018) 6:168 +Page 2 of 18 + +and alkaliphilic genera Desulfonatronobacter (order Desulfo- +bacterales) and Desulfonatronospira (order Desulfovibrio- +nales) +were +the +dominant +Deltaproteobacteria. +Highly +abundant OTUs, within the Actinobacteria belonged to the +class Nitriliruptoria and within the Alphaproteobacteria to +the family Rhodobacteraceae and the genus Roseibaca. The +important nitrifying genus Nitrobacter (Alphaproteobacteria) +was present in only one of the lakes with moderate salinity +(Additional file 3). +Some bacterial top-level taxa appeared less dominant +(< 5%) from the 16S rRNA genes recovered from the +metagenomes but were represented mainly by a single +highly abundant OTU in the amplicon sequences, in- +cluding the haloalkaliphilic genus Truepera within the +phylum Deinococcus-Thermus, the genus Spirochaeata +within the phylum Spirochaetes, the family BSN166 +within the phylum Ignavibacteriae, the BD2–11 terres- +trial group within the Gemmatimonadetes, and the +WCHB1–41 +order +within +the +Verrucomicrobia. +All +OTUs +within +the +Thermotogae +and +Lentisphaerae +belonged to uncultured genera from the family Kosmoto- +gaceae and Oligosphaeraceae, respectively. All Tenericu- +tes-related OTUs belonged to the class Mollicutes, and +especially the order NB1-n was dominant. In contrast, +the phylum Planctomycetes was relatively diverse, with +at least 11 different genus-level OTUs spread over four +class-level groups. +High-throughput genome recovery +We obtained 717 medium-quality (≥ 50% complete, +< 10% contamination) and 154 near-complete (≥ 90% +complete, < 5% contamination) metagenome-assembled +genomes (MAGs) across three major prokaryote groups: +Archaea, Bacteria, and CPR (see Additional file 4 and +Additional file 2: Figure S2). Figures 2 and 3 show the +top-level phylogeny of all MAGs based on 16 ribosomal +proteins. The reference database used contains a repre- +sentative for each major prokaryote lineage [17]. We +a +b +Fig. 1 Abundant prokaryotic groups in five hypersaline soda lake sediments. a Relative abundance of the top-level taxa (those with ≥ 1% abundance +in at least one dataset) based on 16S rRNA reads in unassembled metagenomic datasets. b Relative abundance of the 16S rRNA OTUs (those with sum +of abundance in all datasets ≥ 3%) recovered by amplicon sequencing assigned where possible down to the genus-level. Three of the assessed soda +lakes have a moderate salinity (70–110 g L−1), two are salt-saturated (400 g L− 1) +Vavourakis et al. Microbiome (2018) 6:168 +Page 3 of 18 + +colored the different phyla from which we obtained a +MAG +in +alternate +blue +and +orange +colors, +and +highlighted the MAGs obtained here in a darker shade. +Many MAGs belonged to uncultured groups commonly +detected in soda lake 16S rRNA gene surveys, over 100 +MAGs still belonged to candidate prokaryote phyla and +divisions that to our knowledge were never detected be- +fore in soda lakes, including CPR. Although only few +MAGs had near-complete 16S rRNA genes, in most +cases we were able to link available taxonomic gene an- +notations and ribosomal protein phylogeny to the SILVA +taxonomy of the OTUs assigned to the amplicon se- +quences, while cross-checking the abundance profiles of +both MAGs (Additional file 5) and OTUs. +The soda lake CPR recovered from the metagenomes was +restricted to a few distinct phyla within the Parcubacteria +group, mostly affiliating with “Candidatus Nealsonbacteria” +and “Ca. Zambryskibacteria” [15] (Fig. 2). The first group of +MAGs encompassed four different branches in our riboso- +mal protein tree, suggesting a high-phylogenetic diversity, +with 33 putative new species sampled here (ANI and con- +DNA matrices given in Additional file 6). The “Ca. Zambrys- +kibacteria-”related MAGs consisted of at least five new +species. Few MAGs were recovered from CPR groups also +detected by amplicon sequencing (see Additional file 2: +Figure S1), namely the “Ca. Dojkabacteria” (former WS6), +“Ca. Saccharibacteria” (former TM7), CPR2, and “Ca. +Katanobacteria” (former WWE3). +Fig. 2 Maximum-likelihood phylogeny of the CPR and archaeal MAGs based on 16 ribosomal proteins. The archaeal tree is unrooted. The CPR tree is rooted +to the Wirthbacteria. Alternate orange and blue colors show phyla/classes from which we obtained MAGs (labeled as “Phyla present”). Reconstructed MAGs of +this study are highlighted by darker shades (labeled as “MAG present”). Phyla/classes for which there was no representative in the reconstructed MAGs of this +study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom. Colored circles at the nodes show +confidence percentage of the bootstraps analysis (100×) +Vavourakis et al. Microbiome (2018) 6:168 +Page 4 of 18 + +Most archaeal MAGs belonged to the phylum Euryarch- +aeota and the abundant classes Halobacteria, Methanomi- +crobia, and Thermoplasmata (related to OTU KTK 4A) +within. In addition, three Thermoplasmata-related MAGs +that encoded for the key enzyme for methanogenesis +(methyl-coenzyme M reductase, mcr) affiliated with refer- +ence genomes from Methanomassilicoccales, the seventh +order of methanogens have been recovered [20, 21]. +Another MCR-encoding MAG was closely related to the +latest +discovered +group +of +poly-extremophilic, +methyl-reducing methanogens from hypersaline lakes +from the class Methanonatronarchaeia [9] (related to +OTU ST-12K10A). We recovered also one MAG from the +class Methanobacteria and a high-quality MAG from the +WCHA1–57 +group +(“Candidatus +Methanofastidiosa” +[22]) in the candidate division WSA2 (Arc I). Several +MAGs were recovered from the DPANN archaeal +groups “Ca. Diapherotrites,” “Ca. Aenigmarchaeota,” +(see Additional file 2: Figure S3) and “Ca. Woesearch- +aeota” (former Deep Sea Hydrothermal Vent Group 6, +DHVEG-6). Although we did not reconstruct any +reasonable-sized MAGs from the TACK superphylum, +we found several 16S rRNA genes on the assembled +contigs that affiliated to the Thaumarchaeota (see +Additional file 1: Table S2). +Nearly every known bacterial phylum had an extremo- +philic lineage sampled from our hypersaline soda lake +sediments (Fig. 3). In most cases, the soda lake lineages +clearly formed separate branches appearing as sister +groups to known reference lineages. The highest genome +recovery was from the same top-level taxonomic groups +that were also abundant in our 16S rRNA gene analysis. +From the Verrucomicrobia, most MAGs belonged to the +order WCHB1-41 (16S rRNA gene identity 92–100%). +However, in our ribosomal protein tree, they branched +within the phylum Lentisphaerae. Sixteen Tenericutes +MAGs from at least 12 different species (Additional file 6) +were closely related to the NB1-n group of Mollicutes. +Based on the recovered genome size and encoded meta- +bolic potential, these organisms are free-living anaerobic +fermenters of simple sugars, similar to what has recently +been +proposed +for +“Candidatus +Izimaplasma” +[23]. +Fig. 3 Maximum-likelihood phylogeny of the bacterial MAGs (CPR excluded) based on 16 ribosomal proteins. Alternate orange and blue colors show phyla/ +classes from which we obtained MAGs (labeled as “Phyla present”). Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG +present”). Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not +present”), and the numerical labels are represented at the bottom. Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) +Vavourakis et al. Microbiome (2018) 6:168 +Page 5 of 18 + +Several MAGs belonged to the candidate phyla “Ca. +Omnitrophica,” “Ca. Atribacteria,” and “Ca. Acetother- +mia” (former OP1), which were moderately abundant +also in some sediment (see Additional file 2: Figure S1). +For the latter phylum, we suspect that four MAGs were +more closely related to ca. div. WS1 and “Ca. Lindow- +bacteria” for which only few reference genomes are +currently available in NCBI (see Additional file 2: +Figure S4). Due to a high-sequencing coverage, we also +managed to reconstruct several MAGs from rare Bacteria +(< 100 amplicon sequences detected, see Additional file 2: +Figure S1), including the phyla “Ca. Hydrogenedentes,” +“Ca. Cloacimonetes,” ca. div. BRC1, Elusimicrobia, Caldi- +serica, and “Ca. Latescibacteria.” The MAGs from the +latter phylum were more closely related to the recently +proposed phylum “Ca. Handelsmanbacteria” [15]. Two +additional MAGs with 16S rRNA gene fragments with +94–95% identity to the class MD2898-B26 (Nitrospinae) +were more likely members of ca. div. KSB3 (proposed +“Ca. Moduliflexus” [24], see Additional file 2: Figure S5). +Draft genomes of haloalkaliphilic CPR +Strikingly, members of the CPR related to “Ca. Nealson- +bacteria” and “Ca. Vogelbacteria” were among the top +5% of abundant organisms in the surface sediments of +the soda lakes, especially those with moderate salinity +(Fig. 4). Like most members of the CPR, the MAGs of +the four most abundant “Ca. Nealsonbacteria” seem to +be anaerobic fermenters [25]. They lacked a complete +TCA cycle and most complexes from the oxidative elec- +tron transfer chain, except for the subunit F of a +NADH-quinone oxidoreductase (complex I, nuoF, nuoG, +nuoA) and coxB genes (complex II). All CPR MAGs had +a near-complete glycolysis pathway (Embden-Meyerhof- +Parnas) encoded, but pentose phosphate pathways were +severely truncated. The commonly encoded F- and +V-type ATPase can establish a membrane potential for +symporter-antiporters by utilizing the ATP formed by +substrate-level phosphorylation during fermentation. All +CPR have V-type ATPases that can translocate Na+ in +addition to H+ (see Additional file 2: Figure S6), while +only two members of the “Ca. Falkowbacteria” had puta- +tive Na+-coupled F-type ATPases (see Additional file 2: +Figure S7). The coupling of ATP hydrolysis to sodium +translocation is advantageous to maintain pH homeosta- +sis in alkaline environments. Interestingly, with only two +exceptions [26, 27], all CPR genomes recovered from +other environments with neutral pH were reported to +encode only F-type ATPases [28–32]. One low-abundant +MAG affiliated to “Ca. Peregrinibacteria” contained both +the +large +subunit +of +a +RuBisCO +(type +II/III, +see +Additional file 2: Figure S8) and a putative phosphoribu- +lokinase (PRK, K00855) encoded in the same contig. +This is remarkable because PRK homologs were not +previously identified among CPR, and RuBisCo form II/ +III was inferred to function in a nucleoside salvage path- +way [33]. One “Ca. Saccharibacteria” MAG encoded for +a putative channelrhodopsin (see Additional file 2: +Figure S9). This is the first rhodopsin found among the +CPR and suggests that this enigmatic group of organ- +isms may have acquired evolutionary adaptations to a +life in sunlit surface environments. +A previous study showed that most CPR has coccoid +cell morphotypes with a monoderm cell envelope resem- +bling those from Gram-positives and Archaea but with a +distinct S-layer [34]. Thick peptidoglycans coated with +acidic surface polymers such as teichoic acids help pro- +tect the cells of Gram-positives against reactive hydroxyl +ions in highly alkaline environments [35] (Fig. 5a). All +soda lake CPR had indeed the capability for peptidogly- +can biosynthesis, but we found proteins typical for +Gram-negatives for the biosynthesis of lipopolysaccha- +rides (see Additional file 1: Table S3), homologous to the +inner membrane proteins of type II secretion systems +and +to +several +proteins +associated +to +the +outer +membrane and peptidoglycan, including OmpA. +It remains to be determined whether the soda lake +CPR also lacks an outer membrane and perhaps anchor +lipopolysaccharides, S-layer proteins, and lipoproteins to +the inner cell membrane or peptidoglycan. We also +found gene encoding cardiolipin and squalene synthases. +Increased levels of cardiolipin and the presence of squa- +lene make the cytoplasmic membrane less leaky for +protons [36]. In addition, cation/proton exchangers are +known to play a crucial role for pH homeostasis in alka- +liphilic prokaryotes as they help acidify the cytoplasm +during the extrusion of cations [35]. Putative Na+/H+ +exchangers of the Nha-type and multi-subunit Mnh-type +were found only within a few soda lake CPR. Secondary +active transport of K+ might be mediated in most soda +lake CPR by KefB (COG0475)/kch Kef-type, glutathione- +dependent K+ transport systems, with or without H+ +antiport (67,68). +Various soda lake CPR had an acidic proteome, with +pI curves resembling those found in extremely halophilic +Bacteria. Intracellular proteins enriched in acidic amino +acids might be an adaptation to a “salt-in” strategy, i.e., +maintaining high intracellular potassium (K+) concentra- +tions to keep osmotic balance [7, 37] (Fig. 5b; see +Additional file 2: Figure S10). Such a strategy is energet- +ically favorable over de novo synthesis or import of +osmolytes such as ectoine and glycine betaine. We did +not find genes for the synthesis of organic osmolytes and +homologs of ABC-type transporters for primary active +uptake of proline/glycine betaine which were encoded +only in one MAG (Fig. 5a). For the “Ca. Nealsonbac- +teria” and “Ca. Vogelbacteria,” the salt-in strategy might +be a unique feature for the soda lake species explaining +Vavourakis et al. Microbiome (2018) 6:168 +Page 6 of 18 + +their high abundance in the hypersaline soda lake sedi- +ments, as we did not found an acidic proteome pre- +dicted from genomes obtained from other non-saline +environments (See Additional file 2: Figure S11). The +uptake of K+ ions remains enigmatic for most soda lake +CPR. Low-affinity Trk-type K+ uptake transporters (gen- +erally with symport of H+) (67,68) were encoded only by +a limited number of MAGs. We found three MAGs +Fig. 4 Relative abundance and metabolic potential of the dominant species. Abundance values, expressed as reads per kilobase of MAG per gigabase +of mapped reads (RPKG), were averaged for the top ten abundant MAGs from each dataset that were (likely) the same species (Additional file 5, +Additional file 6). Population genomes were ranked by their “salinity preference scores”: those recruiting relatively more from moderate salinity datasets +(cold colors) are drawn to the top, from high salinity datasets (warm colors) to the bottom. The metabolic potential derived from functional marker +genes (Additional file 7) is depicted by the colored symbols. CBB = Calvin-Benson-Bassham cycle, DNRA = dissimilatory nitrite reduction to ammonia, +fix. = fixation, red. = reduction, ox. = oxidation, dis. = disproportionation +Vavourakis et al. Microbiome (2018) 6:168 +Page 7 of 18 + +a +b +Fig. 5 (See legend on next page.) +Vavourakis et al. Microbiome (2018) 6:168 +Page 8 of 18 + +encoding for Kdp-type sensor kinases (kdpD) but no +corresponding genes for the response regulator (kdpE) +or for Kdp-ATPases that function as the inducible, high- +affinity K+ transporters in other Bacteria (67,68). Finally, +mechanosensitive ion channels (mscS, mscL) and ABC- +type multidrug transport systems (AcrAB, ccmA, EmrA, +MdlB, NorM) and sodium efflux permeases (NatB) were +encoded in almost every MAG. The first might rapidly +restore the turgor pressure under fluctuating salinity +conditions by releasing cytoplasmic ions [38]. +Novel abundant groups involved in sulfur, nitrogen, and +carbon cycles +A new species of Thioalkalivibrio (family Ectothiorhodospir- +aceae) was by far the most abundant in the sediments of +the two salt-saturated lakes (Fig. 4). In the sediment of +Bitter-1, also a purple sulfur bacterium from the same fam- +ily was highly abundant. It was closely related to Halorho- +dospira, a genus also frequently cultured from hypersaline +soda lakes [1]. None of the abundant Ectothiorhodospira- +ceae spp. had already a species-representative genome +sequenced (Additional file 6). The potential of the Thioalk- +alivibrio spp. for chemolithotrophic sulfur oxidation was +evident (Additional file 7; see Additional file 8: Information +S1). Interestingly, the encoded nitrogen metabolisms were +quite versatile. While Thioalkalivibrio sp. 1 had the poten- +tial for nitrate reduction to nitrite, Thioalkalivibrio sp. 2 +might perform dissimilatory nitrite reduction to ammonia +(DNRA; see Additional file 2: Figure S12). +Two +deltaproteobacterial +lineages +of +dissimilatory +sulfate-reducing bacteria (SRB) were highly abundant in +the soda lake sediment of Bitter-1. One MAG from the +family Desulfobacteraceae (order Desulfobacterales) is +the first genome from the genus Desulfonatronobacter. It +encodes the genes for complete sulfate reduction to sul- +fide using various electron donors, as well as for the +complete oxidation of volatile fatty acids and alcohols, a +unique +feature +for +the +genus +Desulfonatronobacter +among haloalkaliphilic SRB [10] (see Additional file 8: +Information S2). Fumarate and nitrite (DNRA, NrfAH) +could be used as alternative electron acceptors. The sec- +ond dominant lineage was a new species from the genus +Desulfonatronospira (family Desulfohalobiaceae, order +Desulfovibrionales). Like other members of this genus, it +had the potential to reduce or disproportionate partially +reduced sulfur compounds. In addition, it could also use +nitrite as an alternative electron acceptor (NrfAH) [6]. +A novel lineage of gammaproteobacterial SOB was +highly abundant in the sediments of the moderately hy- +persaline Cock Soda Lake. It appeared as a sister group of +the family Xanthomonadaceae in the ribosomal protein +tree. This heterotrophic organism could conserve energy +through aerobic respiration. It might detoxify sulfide by +oxidizing it to elemental sulfur (sqr) with subsequent re- +duction or disproportionation of the polysulfides (psrA) +chemically formed from the sulfur. It also encoded the po- +tential for DNRA (nrfA and napC). Genes likely involved +in sulfide detoxification (sqr and psrA) were found also in +several other abundant MAGs of heterotrophs, including +one new abundant species from the family of Nitrilirup- +toraceae (class Nitriliruptoria, phylum Actinobacteria). +We found a wide variety of carbohydrate-active enzymes +in these MAGs, such as cellulases (GH1 family) in +addition to genes for glycolysis and TCA cycle and a +chlorophyll/bacteriochlorophyll a/b synthase (bchG gene). +The latter was also found in other Actinobacteria from the +genus Rubrobacter [39]. No evidence was found for +nitrile-degrading potential. +A second novel, uncultured lineage of Gammaproteo- +bacteria that was highly abundant at moderate salinities +branched in our ribosomal protein tree as a sister group +to the family Halothiobacillaceae. The MAGs encoded +for a versatile metabolism typical for purple non-sulfur +bacteria. The MAGs contained puf genes, bch genes, +genes for carotenoid biosynthesis (not shown), and a +Calvin cycle for photoautotrophic growth. Alternatively, +energy may be conserved through aerobic respiration, +while acetate and proprionate could be taken up via an +acetate permease (actP) and further used for acetyl-CoA +biosynthesis and carbon assimilation. Since the sqr gene +was present, but no dsr or sox genes, the organism +might oxidize sulfide only to elemental sulfur. One bin +contained also nifDKH genes suggesting putative diazo- +trophy, as well as a coenzyme F420 hydrogenase suggest- +ing photoproduction of hydrogen [40]. +The abundant Euryarchaeota organism showed a clear +preference for higher salinities. We obtained one highly +abundant MAG from the class Thermoplasmata that +encoded a full-length 16S rRNA gene only distantly re- +lated (91,2% identity, e value 0) to that of a member of +the KTK 4A group found in a hypersaline endoevaporitic +microbial mat [8]. The abundant soda lake organism is +likely a new genus and species. All KTK 4A-related +MAGs found here encoded for similar heterotrophic, +fermentative +metabolisms, +with +the +potential +for +(See figure on previous page.) +Fig. 5 Potential mechanisms for regulating the intracellular pH and cytoplasmic ion content in different CPR phyla. a Membrane transporters, +channels, and lipids. Peptidoglycan is depicted in gray and S-layer proteins in cyan. b Predicted isoelectric points (bin width 0.2) for the coding +sequences of MAGs. A representative proteome is depicted for each phylum for which several members had a pronounced acidic peak (see also +Additional file 2: Figure S11) +Vavourakis et al. Microbiome (2018) 6:168 +Page 9 of 18 + +anaerobic formate and CO oxidation. The KTK 4A +might be also primary degraders since they encoded pu- +tative cellulases (CAZY-families GH1, GH5) and chiti- +nases (GH18). Interestingly, half of the MAGs encoded a +putative +chlorophyll/bacteriochlorophyll +a/b +synthase +(bchG), which is highly unusual for Archaea. Although +little can be inferred from the presence of only one +marker gene, a functional bchG was previously also +found in Crenarchaeota [41]. The remaining two highly +abundant Euryarchaeota-related MAGs belonged to a +new species of Halorubrum (Additional file 6). +Key genes of the Wood-Ljungdahl pathway found in +novel phylogenetic groups +More than 50 MAGs were related to the family Syntro- +phomonadaceae (class Clostridia, phylum Firmicutes) +based on ribosomal protein phylogeny. All 16S rRNA +gene sequences found in the MAGS had 86–95% iden- +tity to sequences obtained from uncultured organisms +related to the genus Dethiobacter. While an isolated +strain of Dethiobacter alkaliphilus is a facultative auto- +troph +that +respires +thiosulfate, +elemental +sulfur +or +polysulfides with hydrogen as an electron donor [42] or +disproportionates +sulfur +[43], +other +haloalkaliphilic +members +of +the +Syntrophomonadaceae +are +reverse +acetogens, oxidizing acetate in syntrophy with a hydro- +genotrophic partner [44]. Two populations (different +species, Additional file 6) were especially abundant in +Cock Soda Lake (Fig. 4). They encoded for a full +CODH/ACS complex, the key enzyme for the reductive +acetyl-CoA or Wood-Ljungdahl pathway (WL) and a +complete +Eastern +branch +for +CO2 +conversion +to +5-methyl-tetrahydrofolate (Additional file 9) [45, 46]. +Acetogens use the WL to reduce CO2 to acetyl-CoA, +which can be fixed into the cell or used to conserve en- +ergy via acetogenesis. Syntrophic acetate oxidizers, some +sulfate reducing bacteria and aceticlastic methanogens +run the WL in reverse. Syntrophomonadaceae sp. 2 +encoded for a putative thiosulfate/polysulfide reductase +as well as a phosphotransacetylase (pta) and an acetate +kinase (ack) for the ATP-dependent conversion of acet- +ate to acetyl-CoA. Although alternative pathways for the +latter interconversion can exist, this second species has +the complete potential for (reversed) acetogenesis. +Highly remarkable was the presence of a bacterial-type +CODH/ACS +complex +and +a +near-complete +eastern +branch of the WL in a highly abundant species in Cock +Soda Lake from the family Coriobacteriaceae (phylum +Actinobacteria). This prompted us to scan all 871 MAGs +for the presence of acsB encoding for the beta-subunit +of the oxido-reductase module of CODH/ACS. We con- +firmed an encoded +(near)-complete +WL in several +additional organisms belonging to phylogenetic groups +not +previously +associated +with +this +pathway +[46] +(Additional file 9). We removed the Coriobacteriaceae +acsB genes from the final dataset to construct a phylo- +genetic tree since they were < 500 aa (Fig. 6) but found +seven MAGs from the OPB41 class within the Actino- +bacteria (16S rRNA gene fragment identity 94–96%). +The eastern branch of WL can function independently +in folate-dependent C1 metabolism [45], but the pres- +ence of the Western-branch in a phylum that comprises +mostly aerobic isolates is very surprising. The WL in +combination with the potential for acetate to acetyl-CoA +interconversion (pta/ack) and a glycolysis pathway were +also present in the soda lake MAGs from the phyla “Ca. +Handelsmanbacteria,” “Ca. Atribacteria” (latter branched +within the “Ca. Acetothermia”), and the class LD1-PA32 +(Chlamydiae), suggesting all these uncultured organisms +might be heterotrophic acetogens. However, it should be +noted that a PFOR typically connecting glycolysis to the +WL was only encoded in the LD1-PA32 MAGs. More- +over, from the genetic make-up alone, it cannot be +excluded that acetate is activated, and the WL run in +reverse for syntrophic acetate oxidation. Finally, the +novel acsB genes from soda lake Halanaerobiaceae, +Natranaerobiaceae, and Halobacteroidaceae (Firmicutes) +and from Brocadiaceae and Planctomycetaceae (Plancto- +mycetes) disrupt the previously proposed dichotomy +between Terrabacteria and Gracilicutes bacterial groups +unifying 16S rRNA and acsB gene phylogenies [46] and +suggest a far more complex evolutionary history of the +WL pathway than previously anticipated. +Discussion +Extensive +classical +microbiology +efforts +have +been +already undertaken to explore the unique extremophilic +microbial communities inhabiting soda lakes. These un- +covered the presence of most of the functional groups +participating in carbon, nitrogen, sulfur, and minor +element cycling at haloalkaline conditions. The main re- +sults of this work are summarized in several recent re- +views [1, 6, 47, 48]. Since most microbes, including +those living in soda lakes, still evade all cultivation ef- +forts, a very effective way to discover new microbes and +assess their physiology based on their genetic repertoire +is either through single cell genomics or by directly se- +quenced environmental DNA. This exploratory metage- +nomics +study +performed +on +soda +lake +sediments +effectively overcame the existing cultivation bottleneck. +First, we expanded the known diversity of CPR consider- +ably with the first genomes of poly-extremophiles sam- +pled from soda lake sediments. Although the presence of +16S rRNA genes from CPR in marine sediments and hy- +persaline microbial mats was previously shown [34], +until now, CPR MAGs were mainly obtained from deep, +subsurface environments [15, 26, 29, 32, 49–52], and hu- +man microbiota [30]. Despite being highly abundant +Vavourakis et al. Microbiome (2018) 6:168 +Page 10 of 18 + +100 % +90-100 % +70-90 % +50-70 % +some MAGs +all MAGs +Bootstraps +Genes present +Glycolysis (EMP) +PFOR +WL-Eastern branch +H4MPT +TH4 +WL-Western branch +CODH/ACS +Acetogenesis/ +acetate activation +(pta/ack) +0.4 +PVC group (Chlamydiae LD1-PA32) +Syntrophorhabdus aromaticivorans +PVC group bacterium CSSed11_184 +Aerophobetes bacterium SCGC_AAA255-F10 +Ca. Acetothermia +Ca. Handelsmanbacteria +Planctomycetaceae +Anaerolineae +Firmicutes +Brocadiaceae +Planctomycetes +Methanomassiliicoccales +Halobacteroidaceae +Natranaerobiaceae +Methanomicrobiales +Desulfonatronospira +Firmicutes +Dehalococcoidia +Armatimonadetes bacterium CSP1-3 +Deltaproteobacteria +Thermodesulfobacteria +Desulfobulbaceae +Halanaerobiaceae +Nitrospirae +Actinobacteria (OPB41) +Fig. 6 Maximum likelihood phylogeny of the bacterial-type acetyl-coA synthases (acsB) found in the MAGs. Only sequences ≥ 500 aa +were included. Lineages for which we discovered the Wood-Ljungdahl (WL) in this study are highlighted in orange, and the presence +of genes in the respective MAGs related to WL, glycolysis, pyruvate, and acetate conversion is indicated by the colored symbols (see +also Additional file 9: Dataset S6). Additional lineages found in this study are marked in blue. The three was rooted according to [46]. +Circles at the nodes show confidence percentage of the bootstraps analysis (100×). EMP = Embden-Meyerhof-Parnas, PFOR = pyruvate:ferredoxin +oxidoreductase complex, pta = phosphotransacetylase gene, ack = acetate kinase gene, H4MPT = tetrahydromethanopterin-linked pathway, TH4 = +tetrahydrofolate pathway, CODH/ACS = carbon monoxide dehydrogenase/acetyl-CoA synthase. PVC group bacterium CSSed11_184 is likely a member +of the WCHB1-41 class within the Verrucomicrobia +Vavourakis et al. Microbiome (2018) 6:168 +Page 11 of 18 + +here, CPR went unnoticed in previous amplicon sequen- +cing studies. This might be due to the fact that many +CPR representatives have random inserts of various +length in their 16S rRNA genes or due to primer mis- +matches [29, 34]. This illustrates also that direct metage- +nomics should not only be preferred over amplicon +sequencing to infer functional potential, but the former +is far more effective for the discovery of novel organ- +isms. Second, we obtained many more genomes from +“traditional” bacterial phyla such as the Planctomycetes +and Chloroflexi, as well as candidate phyla, for which no +soda lake isolates, hence no genomes were previously +obtained. Third, even within the sulfur cycle, the most +active and frequently studied element cycle in soda lakes +[1], we found considerable metabolic novelty. Finally, we +found the Wood-Ljungdahl pathway in several novel +phyla, not closely related to any known acetogens, +methanogens, or sulfate-reducing bacteria [46]. The lat- +ter shows that our sequencing recovery effort has also +significantly contributed to the discovery of metabolic +novelty within various prokaryote phylogenetic groups. +Salinity is often considered to be the major factor +shaping prokaryote community composition in diverse +habitats [53, 54]. Extreme halophilic Euryarchaeota +seem to be always the dominant group in salt-saturated +hypersaline brines, both those with neutral or alkaline +pH [1, 7, 37]. Here, we found that although these +haloarchaea are still relatively more abundant in the sed- +iments exposed to brines with salt-saturating conditions, +the clear majority of microbes in all investigated hyper- +saline soda lake sediments are Bacteria. It could be +hypothesized that the sediment is a hide-out for the +extreme alkalinity and salinity governing the water +column, and that sediment stratification, especially in +the anoxic part, offers plenty of opportunities for niche +diversification. On the other hand, it should no longer +be a surprise that soda lakes are such productive and +biodiverse +systems. +First, +it +has +been +previously +elaborated that soda lake organisms are exposed to +approximately half the osmotic pressure in sodium +carbonate-dominated +brines +compared +to +sodium +chloride-dominated brines with the same Na+ molarity +[47]. Second, nitrogen limitation in the community can +be overcome when many members contribute to the +fixation of atmospheric N2, and various forms of organic +nitrogen are efficiently recycled. The soda lakes exam- +ined in this study were also eutrophic, and sulfur com- +pounds were abundant. Sulfide is also far less toxic at +high pH as it mostly occurs in the form of bisulfide +(HS−). Besides the evident high metabolic and taxo- +nomic diversity of dissimilatory sulfur-cycling bacteria, a +diverse heterotrophic community can be sustained com- +prising both generalist and very specialized carbon de- +graders. Less eutrophic soda lakes might not suffer from +carbon +limitation +either, +due +to +a +presence +of +high-bicarbonate concentrations. These effectively elim- +inate the inorganic carbon limitation for primary pro- +ducers who are highly active in soda lakes, especially +Cyanobacteria [55, 56]. Third, light that penetrates the +surface of the sediment seems to stimulate oxygenic and +anoxygenic phototrophic growth. Moreover, various het- +erotrophs, such as the rhodopsin-containing haloarchaea +and Bacteroidetes, have the option to tap into this un- +limited energy source for example to help sustain the +costly maintenance of osmotic balance. Unexpectedly, +we even found the first rhodopsin encoded by a member +of the CPR. Fourth, tight syntrophic relations, as pro- +posed for CPR members and Syntrophomonadaceae +spp., might be the solution to successful growth in an +energetically challenging environment. +Since our metagenomes are snapshots in time and space, +the failure to reconstruct specific MAGs gives no conclu- +sive evidence for the absence of certain microbial-mediated +element transformation in hypersaline soda lake sediments. +Additionally, technical limitations of the assembly and bin- +ning of highly micro-diverse genome populations might +hamper genome recovery [57]. More importantly, the +abundance of a specific microbe is not necessarily corre- +lated to the importance of its performance in an ecosystem. +Many metabolic capacities are redundant, and often key +transformations are reserved for a few rare organisms that +might proliferate for a short time-span when specific condi- +tions allow for it. For example, although no MAGs were re- +covered from chemolithoautotrophic nitrifiers [58], we did +detect a Nitrobacter-related OTU by amplicon sequencing +and assembled 16S rRNA genes from Thaumarchaeota, +suggesting bacterial and archaeal nitrifiers are present in +the surface sediments of soda lakes at very low abundance. +Finally, the method of DNA isolation might impact the +community composition apparent in the final metagenome +sequenced. Environmental samples contain complex mix- +tures of different organisms, and it is impossible to find a +protocol where the DNA from every single organism is ex- +tracted as efficiently without compromising the final quality +of the extracted DNA. However, since we find all the im- +portant taxonomic and functional groups known from pre- +vious cultivation-dependent studies back in either our +amplicon sequencing datasets or our directly sequenced +metagenomes, we are confident that the community com- +position and the MAGs presented here are representative +for the microbiomes of the soda lake sediments in the +Kulunda Steppe. +Conclusion +Years of intensive microbiological research on soda lakes +seem to have paid off, since many of the described gen- +era we could detect here have a cultured representative +from soda lakes. However, as many of the abundant +Vavourakis et al. Microbiome (2018) 6:168 +Page 12 of 18 + +lineages and groups found in soda lake sediments are +still uncultured, metagenomics proved to be a helpful +tool to gain primary insights in the potential physiology +and ecology of these poly-extremophilic prokaryotes. +We reconstructed the first genomes for many of such +organisms and proposed new functional roles for the +most abundant ones. Future studies should provide +more in depth analyses of these genomes, especially +from the less abundant organisms that might perform +key ecological processes, such as methanogens and nitri- +fiers. In addition, they should focus on gaining physio- +logical culture-based evidence or proof for in situ +activity for the abundant organisms described here. The +key metabolic insights provided by this metagenomics +study can lead to the design of new cultivation strategies. +In general, sediment communities are far more complex +than those found in the corresponding water column +[53, 59] and are therefore often considered too complex +for efficient metagenomic analysis. Many of the novel +lineages found here may therefore have related neutro- +philic lineages in marine and freshwater sediments that +await discovery. We demonstrate here that, by providing +sufficient sequencing depth, the “state of the art metage- +nomics toolbox” can effectively be used on sediments as +well. +Methods +Site description and sample collection +The top 10 cm sediments from four hypersaline, eutrophic +soda lakes located in the Kulunda Steppe (south-western +Siberia, Altai, Russia) were sampled in July of 2010 and +2011. General features and exact location of the sampled +soda lakes are summarized in Additional file 1: Table S1; a +map of the area was published previously [5]. Cock Soda +Lake (a stand-alone lake, sampled both in 2010 and 2011) +and Tanatar-3 (Tanatar system) were moderately hypersa- +line (~ 100 g L−1) with sandy sediment, while Tanatar-1 +and Bitter-1 (Bitter system) were salt-saturated (400 g L−1) +with sulfide-rich sapropel sediments underlined by rock +trona deposits [7, 60]. Especially, Bitter-1 harbors a very +active microbial community, probably due to its high- +organic and -mineral content. Surface sediments were col- +lected by a plastic corer into sterile glass containers and +transported to the laboratory in a cooler. +DNA isolation, 16S rRNA amplicon, and metagenomic +sequencing +The colloidal fraction of each sediment sample (~ 10% +of 50 g) was separated from the course sandy fraction by +several short (30–60 s) low-speed (1–2,000 rpm in +50 mL Falcon tubes) centrifugation steps and washed +with 1–2 M NaCl solution. The pelleted colloidal sedi- +ment fraction was first subjected to 3 cycles of freezing +in liquid nitrogen/thawing, then re-suspended in 0.1 M +Tris (pH 8)/10 mM EDTA, and then subjected to harsh +bead beating treatment. Next, the samples were incu- +bated with lysozyme (15 mg/mL) for 2 h at 37 °C +followed by a SDS (10% w/v) and proteinase K (10 μg/ +mL) treatment for 30 min. at 45 °C. High molecular +weight DNA was isolated using phenol/chloroform ex- +traction, quality-checked, and sequenced as described +previously [7]. Direct high-throughput sequencing of the +DNA was performed on an Illumina HiSeq 2000 plat- +form to generate 150 b paired-end reads. Amplification +of the V4-V6 region of prokaryote 16S rRNA genes +using barcoded 926F-1392R primers, amplicon purifica- +tion, quantification, and Roche (454)-sequencing was +performed together in a batch with brine samples from +the same sampling campaigns. Barcodes and adapter se- +quences were removed from de-multiplexed amplicon +sequence reads and analyzed with the automated NGS +analysis pipeline of the SILVA rRNA gene database pro- +ject [61] (SILVAngs 1.3, database release version 128) +using default parameters. The OTUs (97% identity) +assigned down to the genus level were only considered +when they had a relative abundance ≥ 0.1% in at least +one of the five datasets. +Processing metagenomics reads, assembly, binning, and +post-binning +Metagenomic raw reads were quality trimmed using +Sickle [62] (version 1.33), and only reads ≥ 21 b were +retained. The prokaryotic community structure at taxo- +nomic top levels was extrapolated from ten million ran- +domly sampled singletons from each dataset. Candidate +16S rRNA fragments > 90 b were identified [63] and +compared against the SILVA SSU database 128 (blastn, +min. length 90, min. identity 80%, e value 1e-5). To ver- +ify that the microbial community composition was in- +deed +mostly +prokaryotic, +we +did +a +more +general +screening of the metagenomics reads that identified also +candidate 18S rRNA fragments > 90 b (see Additional +file 1: Tables S4-S5). The complete trimmed read sets +were assembled into contigs ≥ 1 kb with MEGAHIT [64] +(v1.0.3–6-gc3983f9) using paired-end mode, k min = 21, +k max = 131, k step = 10. Genes were predicted using +Prodigal [65] (v.2.6.2) and RNAs with rna_hmm3 [66] +and tRNAscan-SE [67]. Assembled 16S rRNA sequences +were compared to a manually curated version from the +SILVA SSU database (e value ≥ 1e-5). Predicted protein +sequences +were +annotated +against +KEGG +with +GhostKOALA (genus_prokaryotes + family_eukaryotes ++ viruses) [68]. Marker genes for central metabolic +pathways and key environmental element transforma- +tions were identified based on K number assignments +[15, 69–71]. +Contigs ≥ 2.5 kb were binned with METABAT [72] +(superspecific mode) based on differential coverage +Vavourakis et al. Microbiome (2018) 6:168 +Page 13 of 18 + +values obtained by mapping all five trimmed readsets to +all five contig sets with Bowtie2 [73]. The bins were sub- +jected to post-binning (an overview of the workflow is +given in Additional file 2: Figure S13). Bins were +assessed with lineage-specific single copy genes using +CheckM [74] and further processed with the metage- +nomics workflow in Anvi’o [75] (v2.3.2). Since Candidate +Phyla Radiant (CPR) is not included in the CheckM ref- +erence trees and are likely to have low-genome com- +pleteness, we used an existing training file of 797 CPR +genomes to identify putative CPR bins [76]. Bins with +CheckM-completeness ≥ 50% (884 out of 1778) and an +additional four CPR bins were further processed. Coding +sequences +were +annotated +for +taxonomy +against +NCBI-nr (July, 2017) with USEARCH [77] (5.2.32) to +verify that most hits in each bin were to prokaryotic ref- +erences. Phage or viral contigs were manually removed. +Genome +contamination (redundancy) +was estimated +based on marker sets of universal single copy genes +identified for Bacteria [30] and Archaea [78] as imple- +mented in Anvi’o. Genome coverage was obtained by +mapping trimmed reads with BBMap [79] v36.x (kfilter +31, subfilter 15, maxindel 80). Bins with ≥ 5% redun- +dancy were further refined with Anvi’o using circle phy- +lograms +(guide +trees +tnf-cov: +euclidian +ward) +and +scanned again for CPR. Post-binning resulted in a total +of 2499 metagenome-assembled genomes (MAGs), of +which 871 were either medium-quality genome drafts +(CheckM estimated completeness ≥ 50% and contamin- +ation ≤ 10% [80], Additional file 4) or lower quality draft +genomes from CPR. +Phylogeny of the MAGs was assessed based on 16 +single-copy ribosomal proteins and representative refer- +ence genomes of major prokaryote lineages across the +tree of life [17]. Individual ribosomal proteins in our +MAGs were identified by K number assignments. Only +ribosomal proteins ≥ 80 aa were considered. Initial +maximum-likelihood (ML) trees were constructed to de- +termine which organisms belonged to the Archaea, Bac- +teria, or CPR with FastTree 2 [81] (WAG + CAT). Final +separate trees for the three distant evolutionary groups +were constructed in the same manner. Each ribosomal +protein set was aligned separately with MAFFT [82] +(v7.055b, − auto) and concatenated only if a MAG +encoded at least 8 out of 16 proteins. For all trees, a +100× posterior bootstraps +analysis +was +performed. +Phylogenetic trees were visualized together with gen- +ome statistics and abundance information using iTOL +[83]. We cross-checked the taxonomic assignments +based on the phylogeny of the ribosomal protein cas- +sette +with +the +top +hit +contig annotations +against +NCBI-nr and with the reference lineage obtained with +CheckM. Lastly, we manually corrected the MAGs for +misplaced 16S rRNA genes. The final trees presented +in the manuscript were redrawn using FigTree v1.4.3 +[84]. +Detailed genome analyses +CPR +MAGs +were +re-annotated +more +thoroughly: +genes were predicted with Prokka [85], and functional +predictions were performed by running InterProScan +5 locally on the supplied COG, CDD, TIGRFAMs, +HAMAP, Pfam, and SMART databases [86]. BLAST +Koala was used for KEGG pathway predictions [68]. +To find putative carbohydrate-active enzymes in all +final MAGs, we used the web-resource dbCAN [87] +to annotate all predicted proteins ≥ 80 aa against +CAZy [88]. +To identify the top ten abundant MAGs from each re- +spective dataset, ten million randomly sampled single- +tons were mapped onto each MAG with a cut-off of 95% +identity in minimum of 50 bases. Coverage values were +additionally normalized for genome size and expressed +as reads per kilobase of sequence per gigabase of +mapped reads (RPKG) [89]. A positive score (from 871 +to 1) was assigned to each MAG according to the rank- +ing of the summed RPKG of MAGs in the high-salinity +datasets (B1Sed10 and T1Sed) and a negative score ac- +cording to the ranking of the summed RPKGs in the +moderate salinity datasets (CSSed10, CSSed11, T3Se +d10). Both scores were summed to get a “salinity prefer- +ence score” with MAGs recruiting preferably from high +salinity datasets on the positive end, moderate salinity +datasets in the negative end, and those without prefer- +ence in the middle. +We determined species delineation for the most +abundant MAGs and their closest reference genomes +(NCBI-nr) by Average Nucleotide Identity (ANI) and +conserved DNA-matrices, as follows [90]: ANI ≥ 95%, +conDNA ≥ 69% = same species, ANI ≥ 95%, condDNA +< 69% = might be same species, ANI < 95%, condDNA +< 69% = different species. Single gene trees based on +maximum +likelihood +were +constructed +with +un- +trimmed alignments (MAFFT, L-INS-i model) and +FastTree 2 (WAG + CAT, increased accuracy, -spr4 +-mlacc 2 -slownni) using 100× bootstraps. References +were pulled from eggNOG (v4.5.1) [91] and supple- +mented +with +sequences +from +NCBI-nr +or +refined +according to [7, 33, 46, 92–94]. The curated MAGs +were +scanned +for +the +presence +of +rhodopsin +sequences with the hmmsearch software [95] and a +profile +hidden +Markov +model +(HMM) +of +the +bacteriorhodopsin-like protein family (Pfam accession +number +PF01036). +The +identified +sequences +with +significant similarity were aligned together with a +curated database composed of a collection of type-1 +rhodopsins, using MAFFT (L-INS-i accuracy model) +[82]. This protein alignment was further utilized to +Vavourakis et al. Microbiome (2018) 6:168 +Page 14 of 18 + +construct a maximum likelihood tree with 100× boot- +strap with FastTree 2 [81]. All other genes were +identified using the KEGG annotation. +Additional files +Additional file 1: Table S1. General features of the four sampled soda +lakes at time of sampling. Table S2. SILVA classification of the 16S rRNA +gene sequences found in all ≥1 kb contigs of five soda sediment +metagenomic datasets. Table S3. Enzymes involved in lipopolysaccharide +biosynthesis found among different members of the CPR. Table S4. +Sub-kingdom classification of candidate SSU rRNA gene fragments +found in subsamples of 10 million random forward reads from the +five soda sediment metagenomes. Table S5. Top-level taxonomic +classification of the 18S rRNA gene fragments found in subsamples +of 10 million random forward reads from the five soda sediment +metagenomes. Table S6. Description of the metagenomic datasets, +NCBI Sequence Read Archive (SRA) accession numbers and general +statistics of the assembled contigs. (PDF 740 kb) +Additional file 2: Figure S1. Taxonomic fingerprints determined by 16S +rRNA gene amplicon sequencing. Figure S2. Genome statistics of the +871 MAGs. Figure S3. Phylogeny of MAGs belonging to “Candidatus +Aenigmarchaeota” and “Ca. Nanohaloarchaeota”. Figure S4. Phylogeny of +MAGs related to “Candidatus Acetothermia”, candidate division WS1 and +“Candidatus Lindowbacteria”. Figure S5. Phylogeny of MAGs related to +candidate division KSB3 and “Candidatus Schekmanbacteria”. Figure S6. +Multiple sequence alignment of the V-type ATPase subunits K. Figure S7. +Multiple sequence alignment of the F-type ATPase subunits c. Figure S8. +Maximum likelihood tree of the large subunits of RuBisCo and RubisCo- +like proteins. Figure S9. Maximum likelihood tree of the putative +rhodopsins. Figure S10. Predicted isoelectric points (pI) profiles of all +MAGs from CPR members. Figure S11. Predicted isoelectric points +profiles for members of the “Ca. Nealsonbacteria” and “Ca. Vogelbacteria”. +Figure S12. Multiple sequence alignment of the dissimilatory +cytochrome c nitrite reductases (nrfA/TvNiR, K03385). Figure S13. +Overview of the post-binning workflow used for genome recovery. +(PDF 6548 kb) +Additional file 3: Dataset S1. Relative abundance of the OTUs assigned +to the genus-level within the Archaea, Bacteria and organelles from +Eukaryota detected by 16S rRNA gene amplicon sequencing. The OTUs +with less than 0.1% abundance accross all five datasets are not shown. +The names of highly abundant genera (≥1% in at least one of the data- +sets) are shown in bold. (XLSX 24 kb) +Additional file 4: Dataset S2. Organism names, statistics and general +description incl. Completeness and contamination estimates, phylogeny +and DDBJ/EMBL/Genbank accession numbers of the metagenome +assembled genomes (MAGs) described in this paper. All submitted +versions described in this paper are version XXXX01000000. Size = +recovered genome size, Completeness (Compl1), contamination (Cont), +strain heterogenity (Str het) and Taxon CheckM were inferred from +lineage-specific marker sets and a reference tree build with CheckM [74]. +Additional completeness (compl2) and redundancy (red) estimates were +inferred based on the presence of universal single copy genes for Bacteria +and Archaea [75]. Decision and confidence intervals from the Candidate +Phyla Radiation (CPR) scan [75] are given, as well as the taxonomy of the +besthit in SILVA when 16S rRNA genes were present. Phylum/class 16 +ribosomal proteins is the taxonomy derived from our ribosomal protein +trees (see main text: Figs. 2 and 3). OTU gives the inferred link of a +population genome with our 16S rRNA gene amplicon dataset +(Additional file 3). (XLSX 253 kb) +Additional file 5: Dataset S3. Estimated abundance and derived +salinity preference from each MAG in each metagenomic dataset +expressed as Reads per Kilobase of MAG per Gigabase of mapped reads +(RPKG) and “salinity preference score” (see Methods section), basis for +Fig. 4. (XLSX 143 kb) +Additional file 6: Dataset S4. Average Nucleotide Identity (ANI) and +conserved DNA (condna) matrices to determine species delineation +between the most abundant MAGs shown in Fig. 4, closely related +(less abundant) MAGs and NCBI reference genomes. Decision matrix +shows: 1 = same species, − 1 = might be same species, 0 = different +species (see Methods section). (XLSX 1161 kb) +Additional file 7: Dataset S5. Sheet 1 Presence and absence of marker +genes and putative carbohydrate-active enzymes in the MAGs to infer putative +roles in C, N and S element cycles based on K-number assignments and CAZy +annotations. Sheet 2 Summary basis for Fig. 4. (XLSX 41 kb) +Additional file 8: Information S1. More detailed description of the +main metabolisms encoded by Thioalkalivibrio-related MAGs. +Information S2 More detailed description of the main metabolisms +encoded by Deltaproteobacterial-related MAGs. (PDF 219 kb) +Additional file 9: Dataset 6. Sheet 1 shows the MAGs positive for the +marker gene acsB (K14138) encoding an acetyl-CoA synthase (ACS). The +basis for Fig. 6, namely presence and absence of key genes involved in +the Wood-Ljungdahly pathway, acetogenesis, methanogenesis, glycolysis +and pyruvate to CO2 conversion is shown for each MAG. Sheet 2 shows +the MAGs positive for the marker gene cdhC (K00193) encoding for the +beta subunit of an acetyl-CoA decarboxylase synthase complex. While +acsB and cdhC correspond roughly to the Bacterial-type and Archaeal- +type (methanogens) enzymes with the same function, we found few +discrepancies between marker gene and genome phylogeny within the +Methanomassiliicoccaceae and Chloroflexi. (XLSX 52 kb) +Acknowledgments +We thank Dr. Nikolai Chernych for his technical assistance during the +isolation and purification of metagenomics DNA. We also thank the +Department of Energy Joint Genome Institute for sequencing the +metagenomes. +Funding +CDV and GM were supported by the ERC Advanced Grant PARASOL (no. 322551). +A-SA and RG were supported by the research grant 17-04828S from the Grant +Agency of the Czech Republic. MM was supported by the Czech Academy of +Sciences (Postdoc program PPPLZ application number L200961651). DYS was +supported by the SIAM/Gravitation Program (Dutch Ministry of Education and +Science, grant 24002002) and by the Russian Science Foundation (grant 16–14- +00121). Sequencing was performed by the U.S. Department of Energy Joint +Genome Institute, a DOE Office of Science User Facility, as part of the Community +Sequencing Program (contract no. DE-AC02- 05CH11231). +Availability of data and materials +The raw sequence reads of the five metagenomes have been deposited to +the NCBI Sequence Read Archive (see Additional file 1: Table S6 for accession +numbers and read and contig statistics). The final 871 MAGs described in this +paper have been deposited as Whole Genome Shotgun projects at DDBJ/ +EMBL/GenBank, and accession numbers are listed in Additional file 4 +(BioProject ID PRJNA434545). All versions described in this paper are version +XXXX01000000. The cleaned and dereplicated amplicon sequence datasets +are available in FigShare (https://figshare.com/s/7684627445e3621aba24). +Maximum likelihood trees based on the concatenated alignment of 16 +ribosomal proteins, basis for Figs. 2 and 3, in newick format (.tre file) and +complementary datasets (used to plot completeness, contamination, +genome recovery size, G + C mol% and RPKG in iTOL), as well as K number +assignments for the predicted proteins of all MAGs (KEGG-orthologues, +Ghost Koala) and the fully annotated CPR MAGs supporting the conclusions +of this article are also available in FigShare (https://figshare.com/s/ +7684627445e3621aba24). +Authors’ contributions +GM and DYS initiated this study and were responsible for the fieldwork, +sample preparation, and sequencing effort. CDV conceptualized the research +goals under supervision of DYS and GM, and performed the bioinformatics +analysis under close guidance of A-SA and RG. CDV is the primary author of +this manuscript. MM, RG, and CDV prepared the main figures. All authors +read and approved the final manuscript. +Ethics approval and consent to participate +Not applicable. +Vavourakis et al. Microbiome (2018) 6:168 +Page 15 of 18 + +Consent for publication +Not applicable. +Competing interests +The authors declare that they have no competing interests. +Publisher’s Note +Springer Nature remains neutral with regard to jurisdictional claims in +published maps and institutional affiliations. +Author details +1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, +Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, +University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the +Netherlands. 2Department of Aquatic Microbial Ecology, Institute of +Hydrobiology, Biology Centre CAS, Na Sadkach 7, 370 05 Ceske Budejovice, +Czech Republic. 3Winogradsky Institute of Microbiology, Research Centre of +Biotechnology, Russian Academy of Sciences, 60 let Oktyabrya pr-t, 7, bld. 2, +Moscow, Russian Federation117312. 4Environmental Biotechnology, +Department of Biotechnology, Delft University of Technology, Van der +Maasweg 9, 2629, HZ, Delft, the Netherlands. +Received: 23 June 2018 Accepted: 3 September 2018 +References +1. +Sorokin DY, Berben T, Melton ED, Overmars L, Vavourakis CD, Muyzer G. +Microbial diversity and biogeochemical cycling in soda lakes. 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Microbiome (2018) 6:168 +Page 18 of 18 + diff --git a/kb_25/content/tmp_files/load_file.txt b/kb_25/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a5332718ad78d643b8a6d7d88533c90a2a83843c --- /dev/null +++ b/kb_25/content/tmp_files/load_file.txt @@ -0,0 +1,1142 @@ +filepath=D:\projects\langchain-ChatGLM-master\knowledge_base\kb_25\content\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf,len=1141 +page_content='RESEARCH Open Access A metagenomics roadmap to the uncultured genome diversity in hypersaline soda lake sediments Charlotte D.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Vavourakis1 , Adrian-Stefan Andrei2†, Maliheh Mehrshad2†, Rohit Ghai2, Dimitry Y.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sorokin3,4 and Gerard Muyzer1* Abstract Background: Hypersaline soda lakes are characterized by extreme high soluble carbonate alkalinity.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Despite the high pH and salt content, highly diverse microbial communities are known to be present in soda lake brines but the microbiome of soda lake sediments received much less attention of microbiologists.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Here, we performed metagenomic sequencing on soda lake sediments to give the first extensive overview of the taxonomic diversity found in these complex, extreme environments and to gain novel physiological insights into the most abundant, uncultured prokaryote lineages.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Results: We sequenced five metagenomes obtained from four surface sediments of Siberian soda lakes with a pH 10 and a salt content between 70 and 400 g L−1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The recovered 16S rRNA gene sequences were mostly from Bacteria, even in the salt-saturated lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Most OTUs were assigned to uncultured families.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We reconstructed 871 metagenome-assembled genomes (MAGs) spanning more than 45 phyla and discovered the first extremophilic members of the Candidate Phyla Radiation (CPR).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Five new species of CPR were among the most dominant community members.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Novel dominant lineages were found within previously well-characterized functional groups involved in carbon, sulfur, and nitrogen cycling.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Moreover, key enzymes of the Wood-Ljungdahl pathway were encoded within at least four bacterial phyla never previously associated with this ancient anaerobic pathway for carbon fixation and dissimilation, including the Actinobacteria.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Conclusions: Our first sequencing effort of hypersaline soda lake sediment metagenomes led to two important advances.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' First, we showed the existence and obtained the first genomes of haloalkaliphilic members of the CPR and several hundred other novel prokaryote lineages.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The soda lake CPR is a functionally diverse group, but the most abundant organisms in this study are likely fermenters with a possible role in primary carbon degradation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Second, we found evidence for the presence of the Wood-Ljungdahl pathway in many more taxonomic groups than those encompassing known homo-acetogens, sulfate-reducers, and methanogens.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Since only few environmental metagenomics studies have targeted sediment microbial communities and never to this extent, we expect that our findings are relevant not only for the understanding of haloalkaline environments but can also be used to set targets for future studies on marine and freshwater sediments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Keywords: Soda lake sediments, Metagenomics, Haloalkaliphilic extremophiles, Candidate Phyla Radiation, Wood-Ljungdahl pathway * Correspondence: G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='Muijzer@uva.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='nl †Adrian-Stefan Andrei and Maliheh Mehrshad contributed equally to this work.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the Netherlands Full list of author information is available at the end of the article © The Author(s).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='0 International License (http://creativecommons.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='org/licenses/by/4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The Creative Commons Public Domain Dedication waiver (http://creativecommons.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='org/publicdomain/zero/1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='0/) applies to the data made available in this article, unless otherwise stated.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 https://doi.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='org/10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='1186/s40168-018-0548-7 MicrobiomeBackground Soda lakes are evaporative, athallasic salt lakes with low cal- cium and magnesium concentrations and a high-alkaline pH up to 11 buffered by dissolved (bi-) carbonate ions [1].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' They are constrained to arid regions across the globe, mainly the tropical East African Rift Valley [2], the Libyan Desert [3], the deserts in California and Nevada [4], and the dry steppe belt of Central Asia that spans to southern Si- beria, north-eastern Mongolia, and Inner Mongolia in China [1].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' On top of the extreme salinity and alkaline pH, the Eurasian soda lakes experience extreme seasonal temperature differences, causing highly unstable water re- gimes and fluctuating salinities [5].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Yet, soda lakes harbor diverse communities of haloalkaliphilic microbes, mostly prokaryotes that are well adapted to survive and grow in these extreme environments and consist of similar func- tional groups in soda lakes around the world [1, 2, 6].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The relative abundance of different groups is typically governed by the salinity of the brine [1, 7, 8], and microbial-mediated nutrient cycles become partially hampered only at salt-saturating conditions [1].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' So far, all characterized prokaryotic lineages cultured from soda lakes comprise over 70 different species within more than 30 genera [1, 6, 9, 10].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' From these, only a lim- ited number of genomes have been sequenced today, mostly from chemolithoautotrophic sulfur-oxidizing bac- teria belonging to the genus Thioalkalivibrio (class Gam- maproteobacteria) [1, 11, 12].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' It is well established that metagenomics enables the recovery of genomes and the identification of novel genetic diversity where culturing ef- forts fail [13, 14].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' In recent years, next-generation sequen- cing has recovered a massive number of genomes from previously unknown groups of prokaryotes [15, 16], including a strikingly large and diverse group called “Candidate Phyla Radiation” (CPR), only distantly related to other cultured bacterial lineages [17].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Previously, we conducted a metagenomics study on soda lakes and re- constructed novel genomes from uncultured Bacteroidetes and “Candidatus Nanohaloarchaeaota” living in hypersa- line Siberian soda brines [7].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Here, we turned our atten- tion to the far more complex prokaryotic communities living in the sediments of the hypersaline soda lakes from the same region.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We give a broad overview of all the taxonomic groups sequenced and focus on the metabolic diversity found in the reconstructed genomes of the most abundant, uncultured organisms.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Results Overall prokaryote community structure The salinities from the studied soda lakes ranged from moderately hypersaline (between 70 and 110 g L−1) to salt-saturated (400 g L−1 salt).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The soluble carbonate al- kalinity was in the molar range, and the pH in all lakes was around ten (see Additional file 1: Table S1).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' To give an overview of the overall prokaryotic community com- position in each of the samples, we looked at the taxo- nomic classification of 16S rRNA genes recovered both by amplicon sequencing and direct metagenomics se- quencing (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 1, see also Additional file 2: Figure S1;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Additional file 3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The prokaryotic communities of all five sediment samples were highly diverse and consisted mostly of uncultured taxonomic groups.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Bacteria were more abundant than Archaea, regardless of the salinity of the overlaying brine [7] (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 1).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Euryarchaeota were the second and third largest group in the sediments of the two salt-saturated lakes comprising ~ 10 and ~ 20% of the 16S rRNA genes in the metagenomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Most Euryarchaeota-related OTUs detected by amplicon se- quencing belonged either to the uncultured Thermoplas- mata group KTK 4A (SILVA classification) or the genera Halohasta and Halorubrum (class Halobacteria).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' In ac- cordance with cultivation-dependent studies [6], most OTUs assigned to methanogens were from the class Methanomicrobia, especially the lithotrophic genus Methanocalculus (up to ~ 3%) and the methylotrophic genus Methanosalsum (Additional file 3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The varying ratio of the three dominant bacterial groups, Firmicutes, Bacteroidetes (including the newly proposed phyla Rhodothermaeota and Balneolaeota [18]), and Gammaproteobacteria, showed no clear trend in relation to the salinity in the lakes, but when Firmicutes were domin- ant, Bacteroidetes were less abundant and vice versa.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Most Firmicutes belonged to the order Clostridales.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Uncultured members from the family Syntrophomonadaceae had a relative abundance of more than 5% in all five metagen- omes and comprised in two lakes even ~ 11–20% of the recovered amplicon sequences.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The second most abundant Firmicutes order was Halanaerobiales, particularly the genus Halanaerobium (family Halanaerobiaceae) and un- cultured members of the Halobacteroidaceae.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The majority of Bacteroidetes-related OTUs could not be assigned down to the genus level.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The uncultured ML635J-40 aquatic group (order Bacteroidales) comprised at least 5% of all five prokaryotic communities.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' This group has been previously found to be abundant in Mono Lake [4] (a soda lake) and in an anoxic bioreactor degrading cyanobacterial biomass under haloalkaline conditions [19].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Two other highly abun- dant (up to ~ 8%) uncultured groups from the class Balneo- lia (proposed new phylum Balneolaeota [18]) were also detected in other soda lakes before [3, 4].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Within the Gam- maproteobacteria, the genus Thioalkalivibrio was abundant (~ 3% of the total community), but also uncultured members of HOC36 were prevailing at moderate salinities.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Members of the Deltaproteobacteria, Alphaproteobacteria, and Chloroflexi comprised up to ~ 10% of the detected 16S rRNA gene in some of the metagenomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The GIF9 family of the class Dehalococcoidia was among the top three most abundant OTUs in two lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The extremely salt-tolerant Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 Page 2 of 18 and alkaliphilic genera Desulfonatronobacter (order Desulfo- bacterales) and Desulfonatronospira (order Desulfovibrio- nales) were the dominant Deltaproteobacteria.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Highly abundant OTUs, within the Actinobacteria belonged to the class Nitriliruptoria and within the Alphaproteobacteria to the family Rhodobacteraceae and the genus Roseibaca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The important nitrifying genus Nitrobacter (Alphaproteobacteria) was present in only one of the lakes with moderate salinity (Additional file 3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Some bacterial top-level taxa appeared less dominant (< 5%) from the 16S rRNA genes recovered from the metagenomes but were represented mainly by a single highly abundant OTU in the amplicon sequences, in- cluding the haloalkaliphilic genus Truepera within the phylum Deinococcus-Thermus, the genus Spirochaeata within the phylum Spirochaetes, the family BSN166 within the phylum Ignavibacteriae, the BD2–11 terres- trial group within the Gemmatimonadetes, and the WCHB1–41 order within the Verrucomicrobia.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' All OTUs within the Thermotogae and Lentisphaerae belonged to uncultured genera from the family Kosmoto- gaceae and Oligosphaeraceae, respectively.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' All Tenericu- tes-related OTUs belonged to the class Mollicutes, and especially the order NB1-n was dominant.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' In contrast, the phylum Planctomycetes was relatively diverse, with at least 11 different genus-level OTUs spread over four class-level groups.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' High-throughput genome recovery We obtained 717 medium-quality (≥ 50% complete, < 10% contamination) and 154 near-complete (≥ 90% complete, < 5% contamination) metagenome-assembled genomes (MAGs) across three major prokaryote groups: Archaea, Bacteria, and CPR (see Additional file 4 and Additional file 2: Figure S2).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Figures 2 and 3 show the top-level phylogeny of all MAGs based on 16 ribosomal proteins.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The reference database used contains a repre- sentative for each major prokaryote lineage [17].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We a b Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 1 Abundant prokaryotic groups in five hypersaline soda lake sediments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' a Relative abundance of the top-level taxa (those with ≥ 1% abundance in at least one dataset) based on 16S rRNA reads in unassembled metagenomic datasets.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' b Relative abundance of the 16S rRNA OTUs (those with sum of abundance in all datasets ≥ 3%) recovered by amplicon sequencing assigned where possible down to the genus-level.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Three of the assessed soda lakes have a moderate salinity (70–110 g L−1), two are salt-saturated (400 g L− 1) Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 Page 3 of 18 colored the different phyla from which we obtained a MAG in alternate blue and orange colors, and highlighted the MAGs obtained here in a darker shade.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Many MAGs belonged to uncultured groups commonly detected in soda lake 16S rRNA gene surveys, over 100 MAGs still belonged to candidate prokaryote phyla and divisions that to our knowledge were never detected be- fore in soda lakes, including CPR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Although only few MAGs had near-complete 16S rRNA genes, in most cases we were able to link available taxonomic gene an- notations and ribosomal protein phylogeny to the SILVA taxonomy of the OTUs assigned to the amplicon se- quences, while cross-checking the abundance profiles of both MAGs (Additional file 5) and OTUs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The soda lake CPR recovered from the metagenomes was restricted to a few distinct phyla within the Parcubacteria group, mostly affiliating with “Candidatus Nealsonbacteria” and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Zambryskibacteria” [15] (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The first group of MAGs encompassed four different branches in our riboso- mal protein tree, suggesting a high-phylogenetic diversity, with 33 putative new species sampled here (ANI and con- DNA matrices given in Additional file 6).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Zambrys- kibacteria-”related MAGs consisted of at least five new species.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Few MAGs were recovered from CPR groups also detected by amplicon sequencing (see Additional file 2: Figure S1), namely the “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Dojkabacteria” (former WS6), “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Saccharibacteria” (former TM7), CPR2, and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Katanobacteria” (former WWE3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2 Maximum-likelihood phylogeny of the CPR and archaeal MAGs based on 16 ribosomal proteins.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The archaeal tree is unrooted.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The CPR tree is rooted to the Wirthbacteria.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Alternate orange and blue colors show phyla/classes from which we obtained MAGs (labeled as “Phyla present”).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG present”).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 Page 4 of 18 Most archaeal MAGs belonged to the phylum Euryarch- aeota and the abundant classes Halobacteria, Methanomi- crobia, and Thermoplasmata (related to OTU KTK 4A) within.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' In addition, three Thermoplasmata-related MAGs that encoded for the key enzyme for methanogenesis (methyl-coenzyme M reductase, mcr) affiliated with refer- ence genomes from Methanomassilicoccales, the seventh order of methanogens have been recovered [20, 21].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Another MCR-encoding MAG was closely related to the latest discovered group of poly-extremophilic, methyl-reducing methanogens from hypersaline lakes from the class Methanonatronarchaeia [9] (related to OTU ST-12K10A).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We recovered also one MAG from the class Methanobacteria and a high-quality MAG from the WCHA1–57 group (“Candidatus Methanofastidiosa” [22]) in the candidate division WSA2 (Arc I).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Several MAGs were recovered from the DPANN archaeal groups “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Diapherotrites,” “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Aenigmarchaeota,” (see Additional file 2: Figure S3) and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Woesearch- aeota” (former Deep Sea Hydrothermal Vent Group 6, DHVEG-6).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Although we did not reconstruct any reasonable-sized MAGs from the TACK superphylum, we found several 16S rRNA genes on the assembled contigs that affiliated to the Thaumarchaeota (see Additional file 1: Table S2).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Nearly every known bacterial phylum had an extremo- philic lineage sampled from our hypersaline soda lake sediments (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' In most cases, the soda lake lineages clearly formed separate branches appearing as sister groups to known reference lineages.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The highest genome recovery was from the same top-level taxonomic groups that were also abundant in our 16S rRNA gene analysis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' From the Verrucomicrobia, most MAGs belonged to the order WCHB1-41 (16S rRNA gene identity 92–100%).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' However, in our ribosomal protein tree, they branched within the phylum Lentisphaerae.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sixteen Tenericutes MAGs from at least 12 different species (Additional file 6) were closely related to the NB1-n group of Mollicutes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Based on the recovered genome size and encoded meta- bolic potential, these organisms are free-living anaerobic fermenters of simple sugars, similar to what has recently been proposed for “Candidatus Izimaplasma” [23].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 3 Maximum-likelihood phylogeny of the bacterial MAGs (CPR excluded) based on 16 ribosomal proteins.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Alternate orange and blue colors show phyla/ classes from which we obtained MAGs (labeled as “Phyla present”).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG present”).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 Page 5 of 18 Several MAGs belonged to the candidate phyla “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Omnitrophica,” “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Atribacteria,” and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Acetother- mia” (former OP1), which were moderately abundant also in some sediment (see Additional file 2: Figure S1).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' For the latter phylum, we suspect that four MAGs were more closely related to ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' div.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' WS1 and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Lindow- bacteria” for which only few reference genomes are currently available in NCBI (see Additional file 2: Figure S4).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Due to a high-sequencing coverage, we also managed to reconstruct several MAGs from rare Bacteria (< 100 amplicon sequences detected, see Additional file 2: Figure S1), including the phyla “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Hydrogenedentes,” “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Cloacimonetes,” ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' div.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' BRC1, Elusimicrobia, Caldi- serica, and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Latescibacteria.” The MAGs from the latter phylum were more closely related to the recently proposed phylum “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Handelsmanbacteria” [15].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Two additional MAGs with 16S rRNA gene fragments with 94–95% identity to the class MD2898-B26 (Nitrospinae) were more likely members of ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' div.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' KSB3 (proposed “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Moduliflexus” [24], see Additional file 2: Figure S5).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Draft genomes of haloalkaliphilic CPR Strikingly, members of the CPR related to “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Nealson- bacteria” and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Vogelbacteria” were among the top 5% of abundant organisms in the surface sediments of the soda lakes, especially those with moderate salinity (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 4).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Like most members of the CPR, the MAGs of the four most abundant “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Nealsonbacteria” seem to be anaerobic fermenters [25].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' They lacked a complete TCA cycle and most complexes from the oxidative elec- tron transfer chain, except for the subunit F of a NADH-quinone oxidoreductase (complex I, nuoF, nuoG, nuoA) and coxB genes (complex II).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' All CPR MAGs had a near-complete glycolysis pathway (Embden-Meyerhof- Parnas) encoded, but pentose phosphate pathways were severely truncated.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The commonly encoded F- and V-type ATPase can establish a membrane potential for symporter-antiporters by utilizing the ATP formed by substrate-level phosphorylation during fermentation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' All CPR have V-type ATPases that can translocate Na+ in addition to H+ (see Additional file 2: Figure S6), while only two members of the “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Falkowbacteria” had puta- tive Na+-coupled F-type ATPases (see Additional file 2: Figure S7).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The coupling of ATP hydrolysis to sodium translocation is advantageous to maintain pH homeosta- sis in alkaline environments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Interestingly, with only two exceptions [26, 27], all CPR genomes recovered from other environments with neutral pH were reported to encode only F-type ATPases [28–32].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' One low-abundant MAG affiliated to “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Peregrinibacteria” contained both the large subunit of a RuBisCO (type II/III, see Additional file 2: Figure S8) and a putative phosphoribu- lokinase (PRK, K00855) encoded in the same contig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' This is remarkable because PRK homologs were not previously identified among CPR, and RuBisCo form II/ III was inferred to function in a nucleoside salvage path- way [33].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' One “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Saccharibacteria” MAG encoded for a putative channelrhodopsin (see Additional file 2: Figure S9).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' This is the first rhodopsin found among the CPR and suggests that this enigmatic group of organ- isms may have acquired evolutionary adaptations to a life in sunlit surface environments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' A previous study showed that most CPR has coccoid cell morphotypes with a monoderm cell envelope resem- bling those from Gram-positives and Archaea but with a distinct S-layer [34].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Thick peptidoglycans coated with acidic surface polymers such as teichoic acids help pro- tect the cells of Gram-positives against reactive hydroxyl ions in highly alkaline environments [35] (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 5a).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' All soda lake CPR had indeed the capability for peptidogly- can biosynthesis, but we found proteins typical for Gram-negatives for the biosynthesis of lipopolysaccha- rides (see Additional file 1: Table S3), homologous to the inner membrane proteins of type II secretion systems and to several proteins associated to the outer membrane and peptidoglycan, including OmpA.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' It remains to be determined whether the soda lake CPR also lacks an outer membrane and perhaps anchor lipopolysaccharides, S-layer proteins, and lipoproteins to the inner cell membrane or peptidoglycan.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We also found gene encoding cardiolipin and squalene synthases.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Increased levels of cardiolipin and the presence of squa- lene make the cytoplasmic membrane less leaky for protons [36].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' In addition, cation/proton exchangers are known to play a crucial role for pH homeostasis in alka- liphilic prokaryotes as they help acidify the cytoplasm during the extrusion of cations [35].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Putative Na+/H+ exchangers of the Nha-type and multi-subunit Mnh-type were found only within a few soda lake CPR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Secondary active transport of K+ might be mediated in most soda lake CPR by KefB (COG0475)/kch Kef-type, glutathione- dependent K+ transport systems, with or without H+ antiport (67,68).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Various soda lake CPR had an acidic proteome, with pI curves resembling those found in extremely halophilic Bacteria.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Intracellular proteins enriched in acidic amino acids might be an adaptation to a “salt-in” strategy, i.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='e.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=', maintaining high intracellular potassium (K+) concentra- tions to keep osmotic balance [7, 37] (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 5b;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' see Additional file 2: Figure S10).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Such a strategy is energet- ically favorable over de novo synthesis or import of osmolytes such as ectoine and glycine betaine.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We did not find genes for the synthesis of organic osmolytes and homologs of ABC-type transporters for primary active uptake of proline/glycine betaine which were encoded only in one MAG (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 5a).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' For the “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Nealsonbac- teria” and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Vogelbacteria,” the salt-in strategy might be a unique feature for the soda lake species explaining Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 Page 6 of 18 their high abundance in the hypersaline soda lake sedi- ments, as we did not found an acidic proteome pre- dicted from genomes obtained from other non-saline environments (See Additional file 2: Figure S11).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The uptake of K+ ions remains enigmatic for most soda lake CPR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Low-affinity Trk-type K+ uptake transporters (gen- erally with symport of H+) (67,68) were encoded only by a limited number of MAGs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We found three MAGs Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 4 Relative abundance and metabolic potential of the dominant species.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Abundance values, expressed as reads per kilobase of MAG per gigabase of mapped reads (RPKG), were averaged for the top ten abundant MAGs from each dataset that were (likely) the same species (Additional file 5, Additional file 6).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Population genomes were ranked by their “salinity preference scores”: those recruiting relatively more from moderate salinity datasets (cold colors) are drawn to the top, from high salinity datasets (warm colors) to the bottom.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The metabolic potential derived from functional marker genes (Additional file 7) is depicted by the colored symbols.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' CBB = Calvin-Benson-Bassham cycle, DNRA = dissimilatory nitrite reduction to ammonia, fix.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' = fixation, red.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' = reduction, ox.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' = oxidation, dis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' = disproportionation Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 Page 7 of 18 a b Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 5 (See legend on next page.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=') Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 Page 8 of 18 encoding for Kdp-type sensor kinases (kdpD) but no corresponding genes for the response regulator (kdpE) or for Kdp-ATPases that function as the inducible, high- affinity K+ transporters in other Bacteria (67,68).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Finally, mechanosensitive ion channels (mscS, mscL) and ABC- type multidrug transport systems (AcrAB, ccmA, EmrA, MdlB, NorM) and sodium efflux permeases (NatB) were encoded in almost every MAG.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The first might rapidly restore the turgor pressure under fluctuating salinity conditions by releasing cytoplasmic ions [38].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Novel abundant groups involved in sulfur, nitrogen, and carbon cycles A new species of Thioalkalivibrio (family Ectothiorhodospir- aceae) was by far the most abundant in the sediments of the two salt-saturated lakes (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 4).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' In the sediment of Bitter-1, also a purple sulfur bacterium from the same fam- ily was highly abundant.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' It was closely related to Halorho- dospira, a genus also frequently cultured from hypersaline soda lakes [1].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' None of the abundant Ectothiorhodospira- ceae spp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' had already a species-representative genome sequenced (Additional file 6).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The potential of the Thioalk- alivibrio spp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' for chemolithotrophic sulfur oxidation was evident (Additional file 7;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' see Additional file 8: Information S1).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Interestingly, the encoded nitrogen metabolisms were quite versatile.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' While Thioalkalivibrio sp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 1 had the poten- tial for nitrate reduction to nitrite, Thioalkalivibrio sp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2 might perform dissimilatory nitrite reduction to ammonia (DNRA;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' see Additional file 2: Figure S12).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Two deltaproteobacterial lineages of dissimilatory sulfate-reducing bacteria (SRB) were highly abundant in the soda lake sediment of Bitter-1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' One MAG from the family Desulfobacteraceae (order Desulfobacterales) is the first genome from the genus Desulfonatronobacter.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' It encodes the genes for complete sulfate reduction to sul- fide using various electron donors, as well as for the complete oxidation of volatile fatty acids and alcohols, a unique feature for the genus Desulfonatronobacter among haloalkaliphilic SRB [10] (see Additional file 8: Information S2).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Fumarate and nitrite (DNRA, NrfAH) could be used as alternative electron acceptors.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The sec- ond dominant lineage was a new species from the genus Desulfonatronospira (family Desulfohalobiaceae, order Desulfovibrionales).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Like other members of this genus, it had the potential to reduce or disproportionate partially reduced sulfur compounds.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' In addition, it could also use nitrite as an alternative electron acceptor (NrfAH) [6].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' A novel lineage of gammaproteobacterial SOB was highly abundant in the sediments of the moderately hy- persaline Cock Soda Lake.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' It appeared as a sister group of the family Xanthomonadaceae in the ribosomal protein tree.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' This heterotrophic organism could conserve energy through aerobic respiration.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' It might detoxify sulfide by oxidizing it to elemental sulfur (sqr) with subsequent re- duction or disproportionation of the polysulfides (psrA) chemically formed from the sulfur.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' It also encoded the po- tential for DNRA (nrfA and napC).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Genes likely involved in sulfide detoxification (sqr and psrA) were found also in several other abundant MAGs of heterotrophs, including one new abundant species from the family of Nitrilirup- toraceae (class Nitriliruptoria, phylum Actinobacteria).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We found a wide variety of carbohydrate-active enzymes in these MAGs, such as cellulases (GH1 family) in addition to genes for glycolysis and TCA cycle and a chlorophyll/bacteriochlorophyll a/b synthase (bchG gene).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The latter was also found in other Actinobacteria from the genus Rubrobacter [39].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' No evidence was found for nitrile-degrading potential.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' A second novel, uncultured lineage of Gammaproteo- bacteria that was highly abundant at moderate salinities branched in our ribosomal protein tree as a sister group to the family Halothiobacillaceae.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The MAGs encoded for a versatile metabolism typical for purple non-sulfur bacteria.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The MAGs contained puf genes, bch genes, genes for carotenoid biosynthesis (not shown), and a Calvin cycle for photoautotrophic growth.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Alternatively, energy may be conserved through aerobic respiration, while acetate and proprionate could be taken up via an acetate permease (actP) and further used for acetyl-CoA biosynthesis and carbon assimilation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Since the sqr gene was present, but no dsr or sox genes, the organism might oxidize sulfide only to elemental sulfur.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' One bin contained also nifDKH genes suggesting putative diazo- trophy, as well as a coenzyme F420 hydrogenase suggest- ing photoproduction of hydrogen [40].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The abundant Euryarchaeota organism showed a clear preference for higher salinities.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We obtained one highly abundant MAG from the class Thermoplasmata that encoded a full-length 16S rRNA gene only distantly re- lated (91,2% identity, e value 0) to that of a member of the KTK 4A group found in a hypersaline endoevaporitic microbial mat [8].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The abundant soda lake organism is likely a new genus and species.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' All KTK 4A-related MAGs found here encoded for similar heterotrophic, fermentative metabolisms, with the potential for (See figure on previous page.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=') Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 5 Potential mechanisms for regulating the intracellular pH and cytoplasmic ion content in different CPR phyla.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' a Membrane transporters, channels, and lipids.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Peptidoglycan is depicted in gray and S-layer proteins in cyan.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' b Predicted isoelectric points (bin width 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='2) for the coding sequences of MAGs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' A representative proteome is depicted for each phylum for which several members had a pronounced acidic peak (see also Additional file 2: Figure S11) Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 Page 9 of 18 anaerobic formate and CO oxidation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The KTK 4A might be also primary degraders since they encoded pu- tative cellulases (CAZY-families GH1, GH5) and chiti- nases (GH18).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Interestingly, half of the MAGs encoded a putative chlorophyll/bacteriochlorophyll a/b synthase (bchG), which is highly unusual for Archaea.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Although little can be inferred from the presence of only one marker gene, a functional bchG was previously also found in Crenarchaeota [41].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The remaining two highly abundant Euryarchaeota-related MAGs belonged to a new species of Halorubrum (Additional file 6).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Key genes of the Wood-Ljungdahl pathway found in novel phylogenetic groups More than 50 MAGs were related to the family Syntro- phomonadaceae (class Clostridia, phylum Firmicutes) based on ribosomal protein phylogeny.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' All 16S rRNA gene sequences found in the MAGS had 86–95% iden- tity to sequences obtained from uncultured organisms related to the genus Dethiobacter.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' While an isolated strain of Dethiobacter alkaliphilus is a facultative auto- troph that respires thiosulfate, elemental sulfur or polysulfides with hydrogen as an electron donor [42] or disproportionates sulfur [43], other haloalkaliphilic members of the Syntrophomonadaceae are reverse acetogens, oxidizing acetate in syntrophy with a hydro- genotrophic partner [44].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Two populations (different species, Additional file 6) were especially abundant in Cock Soda Lake (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 4).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' They encoded for a full CODH/ACS complex, the key enzyme for the reductive acetyl-CoA or Wood-Ljungdahl pathway (WL) and a complete Eastern branch for CO2 conversion to 5-methyl-tetrahydrofolate (Additional file 9) [45, 46].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Acetogens use the WL to reduce CO2 to acetyl-CoA, which can be fixed into the cell or used to conserve en- ergy via acetogenesis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Syntrophic acetate oxidizers, some sulfate reducing bacteria and aceticlastic methanogens run the WL in reverse.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Syntrophomonadaceae sp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2 encoded for a putative thiosulfate/polysulfide reductase as well as a phosphotransacetylase (pta) and an acetate kinase (ack) for the ATP-dependent conversion of acet- ate to acetyl-CoA.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Although alternative pathways for the latter interconversion can exist, this second species has the complete potential for (reversed) acetogenesis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Highly remarkable was the presence of a bacterial-type CODH/ACS complex and a near-complete eastern branch of the WL in a highly abundant species in Cock Soda Lake from the family Coriobacteriaceae (phylum Actinobacteria).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' This prompted us to scan all 871 MAGs for the presence of acsB encoding for the beta-subunit of the oxido-reductase module of CODH/ACS.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We con- firmed an encoded (near)-complete WL in several additional organisms belonging to phylogenetic groups not previously associated with this pathway [46] (Additional file 9).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We removed the Coriobacteriaceae acsB genes from the final dataset to construct a phylo- genetic tree since they were < 500 aa (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 6) but found seven MAGs from the OPB41 class within the Actino- bacteria (16S rRNA gene fragment identity 94–96%).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The eastern branch of WL can function independently in folate-dependent C1 metabolism [45], but the pres- ence of the Western-branch in a phylum that comprises mostly aerobic isolates is very surprising.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The WL in combination with the potential for acetate to acetyl-CoA interconversion (pta/ack) and a glycolysis pathway were also present in the soda lake MAGs from the phyla “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Handelsmanbacteria,” “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Atribacteria” (latter branched within the “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Acetothermia”), and the class LD1-PA32 (Chlamydiae), suggesting all these uncultured organisms might be heterotrophic acetogens.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' However, it should be noted that a PFOR typically connecting glycolysis to the WL was only encoded in the LD1-PA32 MAGs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' More- over, from the genetic make-up alone, it cannot be excluded that acetate is activated, and the WL run in reverse for syntrophic acetate oxidation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Finally, the novel acsB genes from soda lake Halanaerobiaceae, Natranaerobiaceae, and Halobacteroidaceae (Firmicutes) and from Brocadiaceae and Planctomycetaceae (Plancto- mycetes) disrupt the previously proposed dichotomy between Terrabacteria and Gracilicutes bacterial groups unifying 16S rRNA and acsB gene phylogenies [46] and suggest a far more complex evolutionary history of the WL pathway than previously anticipated.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Discussion Extensive classical microbiology efforts have been already undertaken to explore the unique extremophilic microbial communities inhabiting soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' These un- covered the presence of most of the functional groups participating in carbon, nitrogen, sulfur, and minor element cycling at haloalkaline conditions.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The main re- sults of this work are summarized in several recent re- views [1, 6, 47, 48].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Since most microbes, including those living in soda lakes, still evade all cultivation ef- forts, a very effective way to discover new microbes and assess their physiology based on their genetic repertoire is either through single cell genomics or by directly se- quenced environmental DNA.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' This exploratory metage- nomics study performed on soda lake sediments effectively overcame the existing cultivation bottleneck.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' First, we expanded the known diversity of CPR consider- ably with the first genomes of poly-extremophiles sam- pled from soda lake sediments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Although the presence of 16S rRNA genes from CPR in marine sediments and hy- persaline microbial mats was previously shown [34], until now, CPR MAGs were mainly obtained from deep, subsurface environments [15, 26, 29, 32, 49–52], and hu- man microbiota [30].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Despite being highly abundant Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 Page 10 of 18 100 % 90-100 % 70-90 % 50-70 % some MAGs all MAGs Bootstraps Genes present Glycolysis (EMP) PFOR WL-Eastern branch H4MPT TH4 WL-Western branch CODH/ACS Acetogenesis/ acetate activation (pta/ack) 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='4 PVC group (Chlamydiae LD1-PA32) Syntrophorhabdus aromaticivorans PVC group bacterium CSSed11_184 Aerophobetes bacterium SCGC_AAA255-F10 Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Acetothermia Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Handelsmanbacteria Planctomycetaceae Anaerolineae Firmicutes Brocadiaceae Planctomycetes Methanomassiliicoccales Halobacteroidaceae Natranaerobiaceae Methanomicrobiales Desulfonatronospira Firmicutes Dehalococcoidia Armatimonadetes bacterium CSP1-3 Deltaproteobacteria Thermodesulfobacteria Desulfobulbaceae Halanaerobiaceae Nitrospirae Actinobacteria (OPB41) Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 6 Maximum likelihood phylogeny of the bacterial-type acetyl-coA synthases (acsB) found in the MAGs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Only sequences ≥ 500 aa were included.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Lineages for which we discovered the Wood-Ljungdahl (WL) in this study are highlighted in orange, and the presence of genes in the respective MAGs related to WL, glycolysis, pyruvate, and acetate conversion is indicated by the colored symbols (see also Additional file 9: Dataset S6).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Additional lineages found in this study are marked in blue.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The three was rooted according to [46].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Circles at the nodes show confidence percentage of the bootstraps analysis (100×).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' EMP = Embden-Meyerhof-Parnas, PFOR = pyruvate:ferredoxin oxidoreductase complex, pta = phosphotransacetylase gene, ack = acetate kinase gene, H4MPT = tetrahydromethanopterin-linked pathway, TH4 = tetrahydrofolate pathway, CODH/ACS = carbon monoxide dehydrogenase/acetyl-CoA synthase.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' PVC group bacterium CSSed11_184 is likely a member of the WCHB1-41 class within the Verrucomicrobia Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 Page 11 of 18 here, CPR went unnoticed in previous amplicon sequen- cing studies.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' This might be due to the fact that many CPR representatives have random inserts of various length in their 16S rRNA genes or due to primer mis- matches [29, 34].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' This illustrates also that direct metage- nomics should not only be preferred over amplicon sequencing to infer functional potential, but the former is far more effective for the discovery of novel organ- isms.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Second, we obtained many more genomes from “traditional” bacterial phyla such as the Planctomycetes and Chloroflexi, as well as candidate phyla, for which no soda lake isolates, hence no genomes were previously obtained.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Third, even within the sulfur cycle, the most active and frequently studied element cycle in soda lakes [1], we found considerable metabolic novelty.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Finally, we found the Wood-Ljungdahl pathway in several novel phyla, not closely related to any known acetogens, methanogens, or sulfate-reducing bacteria [46].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The lat- ter shows that our sequencing recovery effort has also significantly contributed to the discovery of metabolic novelty within various prokaryote phylogenetic groups.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Salinity is often considered to be the major factor shaping prokaryote community composition in diverse habitats [53, 54].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Extreme halophilic Euryarchaeota seem to be always the dominant group in salt-saturated hypersaline brines, both those with neutral or alkaline pH [1, 7, 37].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Here, we found that although these haloarchaea are still relatively more abundant in the sed- iments exposed to brines with salt-saturating conditions, the clear majority of microbes in all investigated hyper- saline soda lake sediments are Bacteria.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' It could be hypothesized that the sediment is a hide-out for the extreme alkalinity and salinity governing the water column, and that sediment stratification, especially in the anoxic part, offers plenty of opportunities for niche diversification.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' On the other hand, it should no longer be a surprise that soda lakes are such productive and biodiverse systems.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' First, it has been previously elaborated that soda lake organisms are exposed to approximately half the osmotic pressure in sodium carbonate-dominated brines compared to sodium chloride-dominated brines with the same Na+ molarity [47].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Second, nitrogen limitation in the community can be overcome when many members contribute to the fixation of atmospheric N2, and various forms of organic nitrogen are efficiently recycled.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The soda lakes exam- ined in this study were also eutrophic, and sulfur com- pounds were abundant.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sulfide is also far less toxic at high pH as it mostly occurs in the form of bisulfide (HS−).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Besides the evident high metabolic and taxo- nomic diversity of dissimilatory sulfur-cycling bacteria, a diverse heterotrophic community can be sustained com- prising both generalist and very specialized carbon de- graders.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Less eutrophic soda lakes might not suffer from carbon limitation either, due to a presence of high-bicarbonate concentrations.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' These effectively elim- inate the inorganic carbon limitation for primary pro- ducers who are highly active in soda lakes, especially Cyanobacteria [55, 56].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Third, light that penetrates the surface of the sediment seems to stimulate oxygenic and anoxygenic phototrophic growth.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Moreover, various het- erotrophs, such as the rhodopsin-containing haloarchaea and Bacteroidetes, have the option to tap into this un- limited energy source for example to help sustain the costly maintenance of osmotic balance.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Unexpectedly, we even found the first rhodopsin encoded by a member of the CPR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Fourth, tight syntrophic relations, as pro- posed for CPR members and Syntrophomonadaceae spp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=', might be the solution to successful growth in an energetically challenging environment.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Since our metagenomes are snapshots in time and space, the failure to reconstruct specific MAGs gives no conclu- sive evidence for the absence of certain microbial-mediated element transformation in hypersaline soda lake sediments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Additionally, technical limitations of the assembly and bin- ning of highly micro-diverse genome populations might hamper genome recovery [57].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' More importantly, the abundance of a specific microbe is not necessarily corre- lated to the importance of its performance in an ecosystem.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Many metabolic capacities are redundant, and often key transformations are reserved for a few rare organisms that might proliferate for a short time-span when specific condi- tions allow for it.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' For example, although no MAGs were re- covered from chemolithoautotrophic nitrifiers [58], we did detect a Nitrobacter-related OTU by amplicon sequencing and assembled 16S rRNA genes from Thaumarchaeota, suggesting bacterial and archaeal nitrifiers are present in the surface sediments of soda lakes at very low abundance.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Finally, the method of DNA isolation might impact the community composition apparent in the final metagenome sequenced.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Environmental samples contain complex mix- tures of different organisms, and it is impossible to find a protocol where the DNA from every single organism is ex- tracted as efficiently without compromising the final quality of the extracted DNA.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' However, since we find all the im- portant taxonomic and functional groups known from pre- vious cultivation-dependent studies back in either our amplicon sequencing datasets or our directly sequenced metagenomes, we are confident that the community com- position and the MAGs presented here are representative for the microbiomes of the soda lake sediments in the Kulunda Steppe.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Conclusion Years of intensive microbiological research on soda lakes seem to have paid off, since many of the described gen- era we could detect here have a cultured representative from soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' However, as many of the abundant Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 Page 12 of 18 lineages and groups found in soda lake sediments are still uncultured, metagenomics proved to be a helpful tool to gain primary insights in the potential physiology and ecology of these poly-extremophilic prokaryotes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We reconstructed the first genomes for many of such organisms and proposed new functional roles for the most abundant ones.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Future studies should provide more in depth analyses of these genomes, especially from the less abundant organisms that might perform key ecological processes, such as methanogens and nitri- fiers.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' In addition, they should focus on gaining physio- logical culture-based evidence or proof for in situ activity for the abundant organisms described here.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The key metabolic insights provided by this metagenomics study can lead to the design of new cultivation strategies.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' In general, sediment communities are far more complex than those found in the corresponding water column [53, 59] and are therefore often considered too complex for efficient metagenomic analysis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Many of the novel lineages found here may therefore have related neutro- philic lineages in marine and freshwater sediments that await discovery.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We demonstrate here that, by providing sufficient sequencing depth, the “state of the art metage- nomics toolbox” can effectively be used on sediments as well.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Methods Site description and sample collection The top 10 cm sediments from four hypersaline, eutrophic soda lakes located in the Kulunda Steppe (south-western Siberia, Altai, Russia) were sampled in July of 2010 and 2011.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' General features and exact location of the sampled soda lakes are summarized in Additional file 1: Table S1;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' a map of the area was published previously [5].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Cock Soda Lake (a stand-alone lake, sampled both in 2010 and 2011) and Tanatar-3 (Tanatar system) were moderately hypersa- line (~ 100 g L−1) with sandy sediment, while Tanatar-1 and Bitter-1 (Bitter system) were salt-saturated (400 g L−1) with sulfide-rich sapropel sediments underlined by rock trona deposits [7, 60].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Especially, Bitter-1 harbors a very active microbial community, probably due to its high- organic and -mineral content.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Surface sediments were col- lected by a plastic corer into sterile glass containers and transported to the laboratory in a cooler.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' DNA isolation, 16S rRNA amplicon, and metagenomic sequencing The colloidal fraction of each sediment sample (~ 10% of 50 g) was separated from the course sandy fraction by several short (30–60 s) low-speed (1–2,000 rpm in 50 mL Falcon tubes) centrifugation steps and washed with 1–2 M NaCl solution.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The pelleted colloidal sedi- ment fraction was first subjected to 3 cycles of freezing in liquid nitrogen/thawing, then re-suspended in 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='1 M Tris (pH 8)/10 mM EDTA, and then subjected to harsh bead beating treatment.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Next, the samples were incu- bated with lysozyme (15 mg/mL) for 2 h at 37 °C followed by a SDS (10% w/v) and proteinase K (10 μg/ mL) treatment for 30 min.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' at 45 °C.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' High molecular weight DNA was isolated using phenol/chloroform ex- traction, quality-checked, and sequenced as described previously [7].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Direct high-throughput sequencing of the DNA was performed on an Illumina HiSeq 2000 plat- form to generate 150 b paired-end reads.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Amplification of the V4-V6 region of prokaryote 16S rRNA genes using barcoded 926F-1392R primers, amplicon purifica- tion, quantification, and Roche (454)-sequencing was performed together in a batch with brine samples from the same sampling campaigns.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Barcodes and adapter se- quences were removed from de-multiplexed amplicon sequence reads and analyzed with the automated NGS analysis pipeline of the SILVA rRNA gene database pro- ject [61] (SILVAngs 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='3, database release version 128) using default parameters.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The OTUs (97% identity) assigned down to the genus level were only considered when they had a relative abundance ≥ 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='1% in at least one of the five datasets.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Processing metagenomics reads, assembly, binning, and post-binning Metagenomic raw reads were quality trimmed using Sickle [62] (version 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='33), and only reads ≥ 21 b were retained.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The prokaryotic community structure at taxo- nomic top levels was extrapolated from ten million ran- domly sampled singletons from each dataset.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Candidate 16S rRNA fragments > 90 b were identified [63] and compared against the SILVA SSU database 128 (blastn, min.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' length 90, min.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' identity 80%, e value 1e-5).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' To ver- ify that the microbial community composition was in- deed mostly prokaryotic, we did a more general screening of the metagenomics reads that identified also candidate 18S rRNA fragments > 90 b (see Additional file 1: Tables S4-S5).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The complete trimmed read sets were assembled into contigs ≥ 1 kb with MEGAHIT [64] (v1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='3–6-gc3983f9) using paired-end mode, k min = 21, k max = 131, k step = 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Genes were predicted using Prodigal [65] (v.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='2) and RNAs with rna_hmm3 [66] and tRNAscan-SE [67].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Assembled 16S rRNA sequences were compared to a manually curated version from the SILVA SSU database (e value ≥ 1e-5).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Predicted protein sequences were annotated against KEGG with GhostKOALA (genus_prokaryotes + family_eukaryotes + viruses) [68].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Marker genes for central metabolic pathways and key environmental element transforma- tions were identified based on K number assignments [15, 69–71].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Contigs ≥ 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='5 kb were binned with METABAT [72] (superspecific mode) based on differential coverage Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 Page 13 of 18 values obtained by mapping all five trimmed readsets to all five contig sets with Bowtie2 [73].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The bins were sub- jected to post-binning (an overview of the workflow is given in Additional file 2: Figure S13).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Bins were assessed with lineage-specific single copy genes using CheckM [74] and further processed with the metage- nomics workflow in Anvi’o [75] (v2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='2).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Since Candidate Phyla Radiant (CPR) is not included in the CheckM ref- erence trees and are likely to have low-genome com- pleteness, we used an existing training file of 797 CPR genomes to identify putative CPR bins [76].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Bins with CheckM-completeness ≥ 50% (884 out of 1778) and an additional four CPR bins were further processed.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Coding sequences were annotated for taxonomy against NCBI-nr (July, 2017) with USEARCH [77] (5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='32) to verify that most hits in each bin were to prokaryotic ref- erences.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Phage or viral contigs were manually removed.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Genome contamination (redundancy) was estimated based on marker sets of universal single copy genes identified for Bacteria [30] and Archaea [78] as imple- mented in Anvi’o.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Genome coverage was obtained by mapping trimmed reads with BBMap [79] v36.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='x (kfilter 31, subfilter 15, maxindel 80).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Bins with ≥ 5% redun- dancy were further refined with Anvi’o using circle phy- lograms (guide trees tnf-cov: euclidian ward) and scanned again for CPR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Post-binning resulted in a total of 2499 metagenome-assembled genomes (MAGs), of which 871 were either medium-quality genome drafts (CheckM estimated completeness ≥ 50% and contamin- ation ≤ 10% [80], Additional file 4) or lower quality draft genomes from CPR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Phylogeny of the MAGs was assessed based on 16 single-copy ribosomal proteins and representative refer- ence genomes of major prokaryote lineages across the tree of life [17].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Individual ribosomal proteins in our MAGs were identified by K number assignments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Only ribosomal proteins ≥ 80 aa were considered.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Initial maximum-likelihood (ML) trees were constructed to de- termine which organisms belonged to the Archaea, Bac- teria, or CPR with FastTree 2 [81] (WAG + CAT).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Final separate trees for the three distant evolutionary groups were constructed in the same manner.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Each ribosomal protein set was aligned separately with MAFFT [82] (v7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='055b, − auto) and concatenated only if a MAG encoded at least 8 out of 16 proteins.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' For all trees, a 100× posterior bootstraps analysis was performed.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Phylogenetic trees were visualized together with gen- ome statistics and abundance information using iTOL [83].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We cross-checked the taxonomic assignments based on the phylogeny of the ribosomal protein cas- sette with the top hit contig annotations against NCBI-nr and with the reference lineage obtained with CheckM.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Lastly, we manually corrected the MAGs for misplaced 16S rRNA genes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The final trees presented in the manuscript were redrawn using FigTree v1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='3 [84].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Detailed genome analyses CPR MAGs were re-annotated more thoroughly: genes were predicted with Prokka [85], and functional predictions were performed by running InterProScan 5 locally on the supplied COG, CDD, TIGRFAMs, HAMAP, Pfam, and SMART databases [86].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' BLAST Koala was used for KEGG pathway predictions [68].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' To find putative carbohydrate-active enzymes in all final MAGs, we used the web-resource dbCAN [87] to annotate all predicted proteins ≥ 80 aa against CAZy [88].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' To identify the top ten abundant MAGs from each re- spective dataset, ten million randomly sampled single- tons were mapped onto each MAG with a cut-off of 95% identity in minimum of 50 bases.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Coverage values were additionally normalized for genome size and expressed as reads per kilobase of sequence per gigabase of mapped reads (RPKG) [89].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' A positive score (from 871 to 1) was assigned to each MAG according to the rank- ing of the summed RPKG of MAGs in the high-salinity datasets (B1Sed10 and T1Sed) and a negative score ac- cording to the ranking of the summed RPKGs in the moderate salinity datasets (CSSed10, CSSed11, T3Se d10).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Both scores were summed to get a “salinity prefer- ence score” with MAGs recruiting preferably from high salinity datasets on the positive end, moderate salinity datasets in the negative end, and those without prefer- ence in the middle.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We determined species delineation for the most abundant MAGs and their closest reference genomes (NCBI-nr) by Average Nucleotide Identity (ANI) and conserved DNA-matrices, as follows [90]: ANI ≥ 95%, conDNA ≥ 69% = same species, ANI ≥ 95%, condDNA < 69% = might be same species, ANI < 95%, condDNA < 69% = different species.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Single gene trees based on maximum likelihood were constructed with un- trimmed alignments (MAFFT, L-INS-i model) and FastTree 2 (WAG + CAT, increased accuracy, -spr4 -mlacc 2 -slownni) using 100× bootstraps.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' References were pulled from eggNOG (v4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='1) [91] and supple- mented with sequences from NCBI-nr or refined according to [7, 33, 46, 92–94].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The curated MAGs were scanned for the presence of rhodopsin sequences with the hmmsearch software [95] and a profile hidden Markov model (HMM) of the bacteriorhodopsin-like protein family (Pfam accession number PF01036).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The identified sequences with significant similarity were aligned together with a curated database composed of a collection of type-1 rhodopsins, using MAFFT (L-INS-i accuracy model) [82].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' This protein alignment was further utilized to Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 Page 14 of 18 construct a maximum likelihood tree with 100× boot- strap with FastTree 2 [81].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' All other genes were identified using the KEGG annotation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Additional files Additional file 1: Table S1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' General features of the four sampled soda lakes at time of sampling.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Table S2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' SILVA classification of the 16S rRNA gene sequences found in all ≥1 kb contigs of five soda sediment metagenomic datasets.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Table S3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Enzymes involved in lipopolysaccharide biosynthesis found among different members of the CPR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Table S4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sub-kingdom classification of candidate SSU rRNA gene fragments found in subsamples of 10 million random forward reads from the five soda sediment metagenomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Table S5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Top-level taxonomic classification of the 18S rRNA gene fragments found in subsamples of 10 million random forward reads from the five soda sediment metagenomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Table S6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Description of the metagenomic datasets, NCBI Sequence Read Archive (SRA) accession numbers and general statistics of the assembled contigs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' (PDF 740 kb) Additional file 2: Figure S1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Taxonomic fingerprints determined by 16S rRNA gene amplicon sequencing.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Figure S2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Genome statistics of the 871 MAGs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Figure S3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Phylogeny of MAGs belonging to “Candidatus Aenigmarchaeota” and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Nanohaloarchaeota”.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Figure S4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Phylogeny of MAGs related to “Candidatus Acetothermia”, candidate division WS1 and “Candidatus Lindowbacteria”.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Figure S5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Phylogeny of MAGs related to candidate division KSB3 and “Candidatus Schekmanbacteria”.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Figure S6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Multiple sequence alignment of the V-type ATPase subunits K.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Figure S7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Multiple sequence alignment of the F-type ATPase subunits c.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Figure S8.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Maximum likelihood tree of the large subunits of RuBisCo and RubisCo- like proteins.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Figure S9.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Maximum likelihood tree of the putative rhodopsins.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Figure S10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Predicted isoelectric points (pI) profiles of all MAGs from CPR members.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Figure S11.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Predicted isoelectric points profiles for members of the “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Nealsonbacteria” and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Vogelbacteria”.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Figure S12.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Multiple sequence alignment of the dissimilatory cytochrome c nitrite reductases (nrfA/TvNiR, K03385).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Figure S13.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Overview of the post-binning workflow used for genome recovery.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' (PDF 6548 kb) Additional file 3: Dataset S1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Relative abundance of the OTUs assigned to the genus-level within the Archaea, Bacteria and organelles from Eukaryota detected by 16S rRNA gene amplicon sequencing.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The OTUs with less than 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='1% abundance accross all five datasets are not shown.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The names of highly abundant genera (≥1% in at least one of the data- sets) are shown in bold.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' (XLSX 24 kb) Additional file 4: Dataset S2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Organism names, statistics and general description incl.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Completeness and contamination estimates, phylogeny and DDBJ/EMBL/Genbank accession numbers of the metagenome assembled genomes (MAGs) described in this paper.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' All submitted versions described in this paper are version XXXX01000000.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Size = recovered genome size, Completeness (Compl1), contamination (Cont), strain heterogenity (Str het) and Taxon CheckM were inferred from lineage-specific marker sets and a reference tree build with CheckM [74].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Additional completeness (compl2) and redundancy (red) estimates were inferred based on the presence of universal single copy genes for Bacteria and Archaea [75].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Decision and confidence intervals from the Candidate Phyla Radiation (CPR) scan [75] are given, as well as the taxonomy of the besthit in SILVA when 16S rRNA genes were present.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Phylum/class 16 ribosomal proteins is the taxonomy derived from our ribosomal protein trees (see main text: Figs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2 and 3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' OTU gives the inferred link of a population genome with our 16S rRNA gene amplicon dataset (Additional file 3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' (XLSX 253 kb) Additional file 5: Dataset S3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Estimated abundance and derived salinity preference from each MAG in each metagenomic dataset expressed as Reads per Kilobase of MAG per Gigabase of mapped reads (RPKG) and “salinity preference score” (see Methods section), basis for Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' (XLSX 143 kb) Additional file 6: Dataset S4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Average Nucleotide Identity (ANI) and conserved DNA (condna) matrices to determine species delineation between the most abundant MAGs shown in Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 4, closely related (less abundant) MAGs and NCBI reference genomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Decision matrix shows: 1 = same species, − 1 = might be same species, 0 = different species (see Methods section).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' (XLSX 1161 kb) Additional file 7: Dataset S5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sheet 1 Presence and absence of marker genes and putative carbohydrate-active enzymes in the MAGs to infer putative roles in C, N and S element cycles based on K-number assignments and CAZy annotations.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sheet 2 Summary basis for Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' (XLSX 41 kb) Additional file 8: Information S1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' More detailed description of the main metabolisms encoded by Thioalkalivibrio-related MAGs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Information S2 More detailed description of the main metabolisms encoded by Deltaproteobacterial-related MAGs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' (PDF 219 kb) Additional file 9: Dataset 6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sheet 1 shows the MAGs positive for the marker gene acsB (K14138) encoding an acetyl-CoA synthase (ACS).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The basis for Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 6, namely presence and absence of key genes involved in the Wood-Ljungdahly pathway, acetogenesis, methanogenesis, glycolysis and pyruvate to CO2 conversion is shown for each MAG.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sheet 2 shows the MAGs positive for the marker gene cdhC (K00193) encoding for the beta subunit of an acetyl-CoA decarboxylase synthase complex.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' While acsB and cdhC correspond roughly to the Bacterial-type and Archaeal- type (methanogens) enzymes with the same function, we found few discrepancies between marker gene and genome phylogeny within the Methanomassiliicoccaceae and Chloroflexi.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' (XLSX 52 kb) Acknowledgments We thank Dr.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Nikolai Chernych for his technical assistance during the isolation and purification of metagenomics DNA.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' We also thank the Department of Energy Joint Genome Institute for sequencing the metagenomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Funding CDV and GM were supported by the ERC Advanced Grant PARASOL (no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 322551).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' A-SA and RG were supported by the research grant 17-04828S from the Grant Agency of the Czech Republic.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' MM was supported by the Czech Academy of Sciences (Postdoc program PPPLZ application number L200961651).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' DYS was supported by the SIAM/Gravitation Program (Dutch Ministry of Education and Science, grant 24002002) and by the Russian Science Foundation (grant 16–14- 00121).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sequencing was performed by the U.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Department of Energy Joint Genome Institute, a DOE Office of Science User Facility, as part of the Community Sequencing Program (contract no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' DE-AC02- 05CH11231).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Availability of data and materials The raw sequence reads of the five metagenomes have been deposited to the NCBI Sequence Read Archive (see Additional file 1: Table S6 for accession numbers and read and contig statistics).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The final 871 MAGs described in this paper have been deposited as Whole Genome Shotgun projects at DDBJ/ EMBL/GenBank, and accession numbers are listed in Additional file 4 (BioProject ID PRJNA434545).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' All versions described in this paper are version XXXX01000000.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' The cleaned and dereplicated amplicon sequence datasets are available in FigShare (https://figshare.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='com/s/7684627445e3621aba24).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Maximum likelihood trees based on the concatenated alignment of 16 ribosomal proteins, basis for Figs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2 and 3, in newick format (.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='tre file) and complementary datasets (used to plot completeness, contamination, genome recovery size, G + C mol% and RPKG in iTOL), as well as K number assignments for the predicted proteins of all MAGs (KEGG-orthologues, Ghost Koala) and the fully annotated CPR MAGs supporting the conclusions of this article are also available in FigShare (https://figshare.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='com/s/ 7684627445e3621aba24).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Authors’ contributions GM and DYS initiated this study and were responsible for the fieldwork, sample preparation, and sequencing effort.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' CDV conceptualized the research goals under supervision of DYS and GM, and performed the bioinformatics analysis under close guidance of A-SA and RG.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' CDV is the primary author of this manuscript.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' MM, RG, and CDV prepared the main figures.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' All authors read and approved the final manuscript.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Ethics approval and consent to participate Not applicable.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbiome (2018) 6:168 Page 15 of 18 Consent for publication Not applicable.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Competing interests The authors declare that they have no competing interests.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Author details 1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the Netherlands.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2Department of Aquatic Microbial Ecology, Institute of Hydrobiology, Biology Centre CAS, Na Sadkach 7, 370 05 Ceske Budejovice, Czech Republic.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 3Winogradsky Institute of Microbiology, Research Centre of Biotechnology, Russian Academy of Sciences, 60 let Oktyabrya pr-t, 7, bld.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2, Moscow, Russian Federation117312.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 4Environmental Biotechnology, Department of Biotechnology, Delft University of Technology, Van der Maasweg 9, 2629, HZ, Delft, the Netherlands.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Received: 23 June 2018 Accepted: 3 September 2018 References 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sorokin DY, Berben T, Melton ED, Overmars L, Vavourakis CD, Muyzer G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microbial diversity and biogeochemical cycling in soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Extremophiles.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2014;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='18:791–809.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Oduor SO, Kotut K.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Soda lakes of the East African Rift System: the past, the present and the future.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' In: Schagerl M, editor.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Soda lakes of East Africa.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Berlin: Springer;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2016.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 365–74.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Mesbah NM, Abou-El-Ela SH, Wiegel J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Novel and unexpected prokaryotic diversity in water and sediments of the alkaline, hypersaline lakes of the Wadi An Natrun, Egypt.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Microb Ecol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2007;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='54:598–617.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Humayoun SB, Bano N, James T, Hollibaugh JT.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Depth distribution of microbial diversity in Mono Lake, a meromictic soda lake in California.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Appl Environ Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2003;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='69:1030–42.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Foti MJ, Sorokin DY, Zacharova EE, Pimenov NV, Kuenen JG, Muyzer G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Bacterial diversity and activity along a salinity gradient in soda lakes of the Kulunda Steppe (Altai, Russia).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Extremophiles.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2008;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='12:133–45.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sorokin DY.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Anaerobic haloalkaliphiles.' metadata={'source': 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'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Parks DH, Rinke C, Chuvochina M, Chaumeil P-A, Woodcroft BJ, Evans PN, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Recovery of nearly 8,000 metagenome-assembled genomes substantially expands the tree of life.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Nat Microbiol.' 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'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Hahnke RL, Meier-Kolthoff JP, García-López M, Mukherjee S, Huntemann M, Ivanova NN, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Genome-based taxonomic classification of Bacteroidetes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Front Microbiol.' metadata={'source': 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'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Nolla-Ardevol V, Strous M, Tegetmeyer HE.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Anaerobic digestion of the microalga Spirulina at extreme alkaline conditions: biogas production, metagenome and metatranscriptome.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Front Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='6:597.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 20.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Borrel G, Parisot N, Harris HM, Peyretaillade E, Gaci N, Tottey W, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Comparative genomics highlights the unique biology of Methanomassiliicoccales, a Thermoplasmatales-related seventh order of methanogenic archaea that encodes pyrrolysine.' metadata={'source': 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'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sorokin DY, Tourova TP, Mußmann M, Muyzer G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Dethiobacter alkaliphilus gen.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' nov.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' sp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' nov.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=', and Desulfurivibrio alkaliphilus gen.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' nov.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' sp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' nov.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=': two novel representatives of reductive sulfur cycle from soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Extremophiles.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2008;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='12:431–9.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 43.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Poser A, Lohmayer R, Vogt C.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Extremophiles KK-, 2013 U.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Disproportionation of elemental sulfur by haloalkaliphilic bacteria from soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Extremophiles.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2013;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='17:1003–12.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 44.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sorokin DY, Abbas B, Tourova TP, Bumazhkin BK, Kolganova TV, Muyzer G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sulfate-dependent acetate oxidation under extremely natron-alkaline conditions by syntrophic associations from hypersaline soda lakes.' metadata={'source': 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'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Ann N Y Acad Sci.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2008;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='1125:129–36.' metadata={'source': 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'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Proc Natl Acad Sci.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2018;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='115:E1166–73.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 47.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Sorokin DY, Banciu HL, Muyzer G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Functional microbiology of soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Curr Opin Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='25:88–96.' metadata={'source': 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'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' In: Schagerl M, editor.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Soda lakes of East Africa.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' Berlin: Springer;' metadata={'source': 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biogeochemical transformations in oil reservoirs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' MBio.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content=' 2016;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_25\\content\\dfa5c766-8ed9-459f-bfba-18ff7fd65b35-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'} +page_content='7:e01669–15.' metadata={'source': 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b/n9AzT4oBgHgl3EQfOPuo/content/tmp_files/2301.01163v1.pdf.txt @@ -0,0 +1,2133 @@ +The study of Kantowski-Sachs perfect fluid cosmological model in +modified gravity +1T. Vinutha, 1K. Niharika and K. Sri Kavya2 +1Dept. of Applied Mathematics, AUCST, Andhra University, Visakhapatnam, India. +2 Dept. of Mathematics, Maharaj Vijayaram Gajapathi Raj College of Engineering, +Vizianagaram-535005, India. +⋆vinuthatummala@gmail.com +Abstract +Kantowski-Sachs perfect fluid cosmological model is explored in modified gravity with +functional form f(R, T)=f1(R)+f2(T) where R is Ricci scalar and T is the trace of energy- +momentum tensor. With this functional form, three different cases have been formulated, +namely negative and positive powers of curvature, logarithmic curvature and exponential +curvature given by f1(R) = R + γR2 − µ4 +R , f1(R) = R + νln(τR) and f1(R) = R + κe−ιR +respectively, and for all these three cases, f2(T) = λT, here γ, λ, µ, ν, τ, κ and ι +are constants. While solving the field equations, two constraints i) Expansion scalar is +proportional to shear scalar ii) Hyperbolic scale factor are used. By using these conditions +the required optimum solutions are obtained. The physical parameters are calculated and +geometrical parameters of three cases are analysed against redshift(z) with the help of +pictorial representation. In the context of f(R, T) gravity energy conditions are discussed +with the help of pressure and energy density. If strong energy condition is positive the +gravity should be attractive but in our model it shows negative it means that cosmic +acceleration is due to antigravity, whereas NEC and DEC are fulfilled. The perturbation +technique is used to test the stability of the background solutions of the obtained models. +The inferences obtained from this paper are in persistent with the present cosmological +observations and the model represents an accelerating universe. +Keywords: Kantowski-Sachs spacetime, f(R, T) theory, perfect fluid. +1 +Introduction +Einstein’s theory of general relativity is the foundation of modern physics and it describes +black holes and gravitational phenomena but it break down to give an explanation of cosmic +acceleration. In recent scenario it is well known that our universe is accelerating [1, 2] and it +is one of the trending topics in cosmology. To understand this mysterious concept, we focused +on dark energy and modified theories of gravity. The universe is going through an accelerated +1 +arXiv:2301.01163v1 [gr-qc] 3 Jan 2023 + +period of expansion and it is revealed by the experiments such as CMBR and SNIa. Dark +energy can be inspected in many ways and reforming the geometric part of the Einstein-Hilbert +action is regarded as the most efficient possible way and these changes lead to so many alter- +native theories of gravity. There are different classes of modified gravity such as f(R) gravity, +f(T) gravity, f(G) gravity, f(R, G) gravity. Among them f(R) gravity has attracted many +researchers because it provides a natural gravitation alternative to dark energy. During the +universe expansion f(R) theory elucidate the change from deceleration phase to acceleration +phase. f(R) theory is presumed to be beneficial for resolution of the hierarchy problem or uni- +fication of grand unified theories with gravity in high energy physics. Sotiriou and Faraoni [3] +have worked on f(R) theories of gravity. Antonio De Felice [4] and Shinji Tsujikawa have +studied on f(R) theories. A new class of f(R, T) gravity presented by Harko et al. [5], by +including trace T in f(R) theory. The T-dependence in f(R, T) gravity may appear from the +presence of imperfect fluids or quantum effects. Among all the modified theories of gravitation, +the f(R, T) theory is a generalized theory because there is an energy transfer relation between +matter and geometry. The existence of this relationship is the cause of the rapid expansion of +the universe. The authors who worked on f(R, T) gravity are included in references [6–13]. +In this paper, we examine three specific cases one of them is combination of +1 +Rx and Ry i.e. +f(R) = R+γRy − µ4 +Rx where γ and µ are constants. In this functional form, it has both positive +and negative curvature powers. At low curvature it leads to gravitational alternative for dark +energy which helps in speed up of cosmic expansion where as high curvature describes the infla- +tionary stage of early universe [14]. By considering Ry term for 1 < y < 2 power law inflation +happen at early stage. If y = 2, Starobinsky inflation takes place [15], the term R2 indicates nat- +ural correction to general relativity. According to Nojiri and Odinstov [16] R2 term is necessary +to get rid of instabilities, linear growth of the gravitational force, produce early time inflation +and appear to pass the solar system tests. The state of no linear growth for gravitational force +makes it very much fascinating. Higher derivative terms like R2, R3 can be used to put down +the instabilities significantly. For equivalent scalar tensor theory the solar system test may +be passed as scalar has large mass originated again by higher derivative terms. The standard +Einstein’s gravity may be modified by considering a 1 +R term in the Einstein Hilbert action [17] +which represents the present acceleration of the universe. But the insertion of +1 +R term gener- +ates instabilities which can be overcome by addition of R2 term to the Einstein’s gravitational +action. Besides the advantages of this functional form, have well acceptable Newtonian limit, +no instabilities and no Brans Dicke problem in scalar tensor version. When we put y = 2 and +x = 1 in the above functional form f(R) = R+γRy − µ4 +Rx it reduces to f(R) = R+γR2− µ4 +R and +the obtained results are very efficient. In addition to this functional form by using the linear +function of f(T) = λT, we get the final form of f(R, T) = R + γR2 − µ4 +R + λT where γ, µ and λ +2 + +are constants. Vinutha et al. [18] have worked on Kantowski–Sachs perfect fluid cosmological +model in R2- Gravity. Vinutha and Sri Kavya [19] have studied Bianchi type cosmological mod- +els in f(R, T) theory with quadratic functional form. Brookfield [20] have worked on viability +of f(R) theories with additional powers of curvature. Godani and Samanta [21] have studied +traversable warmholes on f(R) gravity where f(R) = R+αRn. Banik et al. [22] have discussed +Bianchi-I cosmological model in f(R) = R − +β +Rn gravity. +Next, we consider logarithmic curvature i.e. f(R, T) = R + νln(τR) + λT where τ, ν and λ +are constants. As this modified gravity has put forward a gravitational alternative for dark en- +ergy, it is quite interesting to work on this particular functional form. In this model logarithmic +terms are produced by quantam effects in curved space time. The need for dark energy may be +eradicated by this modified gravity and may aid for the fusion of the early time inflation and +cosmic acceleration. Nojiri and Odinstov have studied about modified gravity and proposed +some functional forms such as ln(R) or R−n(lnR)m and R + γR−n(ln R +µ2)m [23,24]. Fayyaz and +Shamir [25] have analysed wormhole structures in logarithmic-corrected R2 gravity. Kourosh +and Tahereh [26] have discussed phantom-like behavior in f(R) = R + βlog( R +µ2) + γRm gravity. +By appending the torsion scalar component to the exponential f(R) theory [27–31], the +functional form is f(R, T) = R + κe−ιR + λT where κ, ι and λ are constants. The reason +behind choosing this functional form it comes up with the best way of exploring cosmic acceler- +ation. In contrast to the ΛCDM model the exponential gravity model has one more parameter +included in it and it also permits the relaxation of fine tuning. Vinutha et al. [32] have stud- +ied on Bianchi type cosmological models in modified theory with exponential functional form. +Paul et al. [33] have worked on accelerating universe in modified theories of gravity. Sahoo +et al. [34] have studied on f(R, T) = f(R) + λT gravity models as alternatives to cosmic ac- +celeration. Moreas and Sahoo [35] have discussed traversable wormholes by using functional +form f(R, T) = R + γeχT and also with this functional form Moreas et al. [36] studied FRW +cosmological model. +When compared to other anisotropic metrics, Kantowski-Sachs model is very simple and +easy to analyze. The cosmologies of Kantowski-Sachs metric possess two properties of symme- +try such as spherical symmetry and invariance under spatial translation. It describes spatially +homogeneous, anisotropic universe and interior of black holes that does not allow a simply +transitive group of motions. It is also used to analyze the behavior of the added degrees of +freedom in quantum cosmological model. This metric represents three different anisotropic +3 + 1 dimensional space time and positive curvature models. The study of anisotropic models +were nourished by the theoretical studies and observations of CMB which also been extended +to modified theories of gravity. Thus this model with an anisotropic nature appeared most +appropriate in describing the early stage of the cosmos. some of the authors who worked on +3 + +Kantowski Sachs model are [37–43]. +This article is organized as follows: In section 2, f(R, T) gravity field equations are obtained +and in section 3 the field equations of power-law, logarithmic and exponential functional forms +are solved. Section 4 discusses the physical and geometrical properties of three cases using +graphs and section 5 concludes our results. +2 +A brief review of f(R, T) = f1(R) + f2(T) model +The final action principle of f(R, T) gravity which is a function of matter Lagrangian Lm is +read as +S = +� � 1 +16πf(R, T) + Lm +� √−gd4x, +(1) +where g is the metric determinant of the fundamental tensor gij, f(R, T) is an arbitrary function +of R and T which is mentioned in the abstract, Lm is the usual matter Lagrangian density and +we consider G = c = 1. +By varying the above equation (1) with respect to gij,we obtain the field equations of f(R, T) +gravity in covariant tensor form as +fR(R, T)Rij − 1 +2f(R, T)gij + (gij□ − ∇i∇j)fR(R, T) = 8πTij − fT(R, T)θij − fT(R, T)Tij, (2) +here, ∇i is the covariant derivative and □ = ∇i∇j is the D’Alemberts operator. fR = ∂f(R,T) +∂R +, +fT = ∂f(R,T) +∂T +and Rij is the Ricci tensor, where +θij = −2Tij + gijLm − 2glk ∂2Lm +∂gijglk . +(3) +Here the energy-momentum tensor is considered to be a perfect fluid which is defined as +Tij = (p + ρ)uiuj − pgij, +(4) +where ui denotes four velocity vector in co-moving coordinates i.e. ui = (1, 0, 0, 0) and uiuj = 1. +Hence, the components of energy-momentum tensor become Tij=diag(ρ, −p, −p, −p), where p +is the pressure and ρ is the energy density of perfect fluid. Several authors have studied by +choosing energy-momentum tensor as perfect fluid which are included in the references [44–51]. +It takes the form by replacing matter Lagrangian as Lm = −p [52–54] in equation (3). +θij = −2Tij − pgij +(5) +4 + +Consequently the field equations for f(R, T) gravity are procured with the aid of T = ρ − 3p +in equation (2) as +Gij = +1 +fR(R, T) +� +[8π + fT(R, T)]Tij + pfT(R, T)gij + 1 +2[f(R, T) − RfR(R, T)]gij +− (gij□ − ∇i∇j)fR(R, T) +� +, +(6) +where Gij is the Einstein tensor which is expressed as Rij − 1 +2Rgij. +Here, we consider the functional form f(R, T) = f1(R) + f2(T) i.e. +f(R, T) = R + γR2 − µ4 +R + λT +(7) +f(R, T) = R + νln(τR) + λT +(8) +f(R, T) = R + κe−ιR + λT +(9) +as case I, II and III respectively. +3 +Metric and solutions of the field equations +Now the metric takes the form, +ds2 = dt2 − M 2(t)dr2 − N 2(t)(dθ2 + sin2 θdψ2, ) +(10) +where M and N are metric potentials and functions of cosmic time t only and co-moving coor- +dinates are (r, θ, ψ). +3.1 +Case I - (negative and positive powers of curvature) +The functional form f(R, T) = R + γR2 − µ4 +R + λT field equations are as follows +1 +N 2 + 2 ¨N +N + +˙N 2 +N 2 = − (8π + 3λ +2 )p +1 + 2Rγ + µ4 +R2 ++ +λρ +2(1 + 2Rγ + µ4 +R2) +− +( γR2 +2 + µ4 +R ) +1 + 2Rγ + µ4 +R2 +− +(2γ − 2µ4 +R3 ) +1 + 2Rγ + µ4 +R2 +�2 ˙N +N +˙R + ¨R +� +− +6µ4 ˙R2 +R4 +1 + 2Rγ + µ4 +R2 +. +� +� +� +� +� +� +� +� +� +� +� +(11) +¨ +M +M + +¨N +N + +˙M ˙N +MN = − (8π + 3λ +2 )p +1 + 2Rγ + µ4 +R2 ++ +λρ +2(1 + 2Rγ + µ4 +R2) +− +( γR2 +2 + µ4 +R ) +1 + 2Rγ + µ4 +R2 +− +(2γ − 2 µ4 +R3) +1 + 2Rγ + µ4 +R2 +� +( +˙M +M + +˙N +N ) ˙R + ¨R +� +− +6 µ4 ˙R2 +R4 +1 + 2Rγ + µ4 +R2 +. +� +� +� +� +� +� +� +� +� +� +� +(12) +5 + +2 +˙M ˙N +MN + +˙N 2 +N 2 + 1 +N 2 = +(8π + 3λ +2 )ρ +1 + 2Rγ + µ4 +R2 +− +λp +2(1 + 2Rγ + µ4 +R2) +− +( γR2 +2 + µ4 +R ) +1 + 2Rγ + µ4 +R2 +− +(2γ − 2 µ4 +R3) +1 + 2Rγ + µ4 +R2 +� ˙M +M + 2 ˙N +N +� +˙R. +� +� +� +� +� +� +� +� +� +� +� +(13) +here dot denotes derivate with respect to t. +3.2 +Case - II (logarithmic curvature) +Field equations corresponding to the f(R, T) = R + νln(τR) + λT are +1 +N 2 + 2 ¨N +N + +˙N 2 +N 2 = −(8π + 3λ +2 )p +1 + ν +R ++ +λρ +2(1 + ν +R) − ν(1 − ln(τR)) +2(1 + ν +R) ++ +�2 ˙N +N +˙R + ¨R +� +ν +R2 +1 + ν +R +− +2ν ˙R2 +R3 +1 + ν +R +. +� +� +� +� +� +� +� +� +� +(14) +¨ +M +M + +¨N +N + +˙M ˙N +MN = −(8π + 3λ +2 )p +1 + ν +R ++ +λρ +2(1 + ν +R) − ν(1 − ln(τR)) +2(1 + ν +R) ++ +ν +R2 +1 + ν +R +� +( +˙M +M + +˙N +N ) ˙R + ¨R +� +− +2ν ˙R2 +R3 +1 + ν +R +. +� +� +� +� +� +� +� +� +� +(15) +2 +˙M ˙N +MN + +˙N 2 +N 2 + 1 +N 2 = (8π + 3λ +2 )ρ +1 + ν +R +− +λp +2(1 + ν +R) − ν(1 − ln(τR)) +2(1 + ν +R) ++ +ν +R2 +1 + ν +R +� ˙M +M + 2 ˙N +N +� +˙R. +� +� +� +� +� +� +� +� +� +(16) +3.3 +Case - III (exponential curvature) +Field equations corresponding to the f(R, T) = R + κe−ιR + λT are given as follows: +1 +N 2 + 2 ¨N +N + +˙N 2 +N 2 = −(8π + 3λ +2 )p +1 − κιe−ιR + +λρ +2(1 − κιe−ιR) + κe−ιR(1 + ιR) +2(1 − κιe−ιR) +− +κι2e−ιR +1 − κιe−ιR +� +(2 ˙N +N ) ˙R + ¨R +� ++ κι3e−ιR ˙R2 +1 − κιe−ιR. +� +� +� +� +� +� +� +� +� +(17) +¨ +M +M + +¨N +N + +˙M ˙N +MN = −(8π + 3λ +2 )p +1 − κιe−ιR + +λρ +2(1 − κιe−ιR) + κe−ιR(1 + ιR) +2(1 − κιe−ιR) +− +κι2e−ιR +1 − κιe−ιR +� +( +˙M +M + +˙N +N ) ˙R + ¨R +� ++ κι3e−ιR ˙R2 +1 − κιe−ιR. +� +� +� +� +� +� +� +� +� +(18) +6 + +2 +˙M ˙N +MN + +˙N 2 +N 2 + 1 +N 2 = (8π + 3λ +2 )ρ +1 − κιe−ιR − +λp +2(1 − κιe−ιR) + κe−ιR(1 + ιR) +2(1 − κιe−ιR) +− κι2e−ιR ˙R +1 − κιe−ιR +� ˙M +M + 2 ˙N +N +� +. +� +� +� +� +� +� +� +� +� +(19) +To obtain solutions for highly non-linear equations is very strenuous and in order to remove +such complications we require some constraints. +(i) We consider σ is proportional to θ(where σ is the shear scalar and θ is the expansion scalar) +and it generate linear relationship between two metric potentials in terms of M and N as +M = N n, +(20) +n ̸= 0, 1 is constant. +The physical motivation for assuming this condition is that Hubble +expansion of the universe may attain isotropy by the observations of the velocity redshift +relation for extra galatic sources if the value of σ +θ is constant [55]. +(ii) The average scale factor is assumed as a hyperbolic expansion +a(t) = sinh(αt) +1 +β , +(21) +where α > 0, β > 0 are constants. The consequence of using this scale factor is time dependent +deceleration parameter q [56]. This average scale factor tends to zero if t → 0 and if t → ∞ +then a(t) becomes infinity. +The directional Hubble parameters are +H1 = +˙M +M +H2 = H3 = +˙N +N . +(22) +The average Hubble parameter is, +H = 1 +3(H1 + 2H2). +(23) +By substituting equation (23) in equation (22), we get +H = ˙a +a = 1 +3 +� ˙M +M + +˙N +N +� +. +(24) +From equations (20) - (24), we obtain metric potentials of M and N as +M = (sinh(αt)) +3n +β(n+2), +(25) +N = (sinh(αt)) +3 +β(n+2). +(26) +If t → ∞ then M and N are nonzero, hence, our model is free from singularity. +Using equations (25) and (26), the Kantowski sachs metric obtained as +ds2 = dt2 − (sinh(αt)) +6n +β(n+2) dr2 − (sinh(αt)) +6 +β(n+2)(dθ2 + sin2 θdψ2). +(27) +The above metric represents a perfect fluid Kantowski-Sachs universe in f(R, T) theory of grav- +ity. +7 + +3.4 +Pressure and energy density for case I +By solving the equations of (11),(12) and (13) we get the pressure of the model as +p = 1 +4 +�χ + ξ − 2η − φ4 + 2φ5 + 2φ6 − φ7 +φ2 − φ1 +− χ + ξ + 2η + 4φ3 + 7φ4 + 2φ5 + 2φ6 + 3φ7 +φ1 + φ2 +� +, +(28) +and the energy density of the model is obtained as +ρ = 1 +4 +�χ + ξ + 2η + 4φ3 + 7φ4 + 2φ5 + 2φ6 + 3φ7 +φ1 + φ2 ++ χ + ξ − 2η − φ4 + 2φ5 + 2φ6 − φ7 +φ2 − φ1 +� +, +(29) +3.5 +Pressure and energy density for case II +By solving the equations (14),(15) and (16) we get the expression for pressure is +p = 1 +4 +�χ + ξ − 2η + φ4 − 2φ5 + 4φ6 + φ7 +φ2 − φ1 +− χ + ξ + 2η + 4φ3 − 7φ4 − 2φ5 + 4φ6 − 3φ7 +φ1 + φ2 +� +, +(30) +and the energy density of the model is obtained as +ρ = 1 +4 +�χ + ξ + 2η + 4φ3 − 7φ4 − 2φ5 + 4φ6 − 3φ7 +φ1 + φ2 ++ χ + ξ − 2η + φ4 − 2φ5 + 4φ6 + φ7 +φ2 − φ1 +� +, +(31) +3.6 +Pressure and energy density for case III +By solving the equations (17), (18) and(19) we get the pressure of the model as +p = 1 +4 +�χ + ξ − 2η − φ4 + 2φ5 − 2φ6 − φ7 +φ2 − φ1 +− χ + ξ + 2η − 4φ3 + 7φ4 + 2φ5 − 2φ6 + 3φ7 +φ1 + φ2 +� +, +(32) +and the energy density of the model is obtained as +ρ = 1 +4 +�χ + ξ + 2η − 4φ3 + 7φ4 + 2φ5 − 2φ6 + 3φ7 +φ1 + φ2 ++ χ + ξ − 2η − φ4 + 2φ5 − 2φ6 − φ7 +φ2 − φ1 +� +, +(33) +The values of χ, ξ and η are same for all the three cases whereas, the values of φi, for i=1 to 7 +for the corresponding cases are clearly given in appendix section. +4 +Physical and geometrical properties +The average Hubble parameter is +H = α +β coth(αt). +(34) +8 + +From the figure of Hubble parameter, we trace that it decreases with the decrease of redshift +i.e. decreases as time increases. By choosing the values of α = 0.21 and β = 3.10 in the scale +factor the Hubble parameter is obtained as 0.07Gyrs−1 which is nearly equal to the present +observational data [57]. For this quantity the dimension is +1 +time. By using this formula, we can +also measure the age of the cosmos. +(ii) The volume of the model is given by +Figure 1: Plot of Hubble parameter(H) versus redshift(z) +V = a3 = (sinh(αt)) +3 +β . +(35) +In figure 2, it is clear that the spatial volume increases with the decrease of redshift i.e. it +increases as the time increases and is finite at final epoch. +(iii) The expansion scalar θ is +Figure 2: Plot of volume(V ) versus redshift(z) +θ = ui +;i = 3H = 3α coth(αt) +β +. +(36) +From figure 3, it is observed that expansion scalar decreases with the decrease of redshift i.e. +it decreases as time increases. Here we noticed that for t = 0 the expansion scalar is infinite. +(iv) We get the shear scalar as +9 + +0.25 +Hubble parameter(H) +0.2 +0.15 +0.1 +0.05 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)30 +25 +Volume(V) +20 +15 +10 +5 +0 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)Figure 3: Plot of expansion scalar(θ) versus redshift(z) +σ2 = 3α2(n − 1)2 coth2(αt) +β2(n + 2)2 +, +(37) +when t = 0, σ2 (shear scalar) tends to infinity. +(v) The mean anisotropy parameter Ah is obtained as +Ah = 1 +3 +� +3 +� +i=1 +�Hi − H +H +�2� +(38) +where i = 1, 2, 3 indicate the directional Hubble parameters for the coordinates of r, θ and ψ. +The mean anisotropy parameter is defined on the basis of directional Hubble parameter and +mean Hubble parameter. +Ah = 2(n − 1)2 +(n + 2)2 ; n ̸= −2. +(39) +The mean anisotropy parameter Ah is useful for checking if the model is anisotropic or not. In +the present model Ah = 0 for n = 1 and Ah ̸= 0 for n ̸= 1 that is the model is anisotropic for +n ̸= 1 and isotropic for n = 1. +In all the discussions and graphical representation of physical parameters we constraint the +constants for case I as α = 0.21, β = 3.10, n = 7.38, λ = −10.02, µ = 0.2, γ = 0.03, case II as +ν = 0.001, τ = 0.002 and case III as κ = 0.2, ι = 0.009. The values of parameters α, β, n, λ in +cases II and III are same as that of Case I. +(iv) The deceleration parameter is +q = −1 + d +dt( 1 +H ). +(40) +In this model by using hyperbolic function we obtained deceleration parameter as +q = −1 + β(1 − tanh2(αt)). +(41) +When t < 1 +α tanh−1(1 − 1 +β) +1 +2, q has negative value which represents that the universe is acceler- +ating whereas if t > 1 +α tanh−1(1− 1 +β) +1 +2, q has positive value which represents that the universe is +10 + +0.8 +expansion scalar(0) +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)decelerating. The quantities such as q and H specifies the geometric properties of the cosmos. +v)Throughout the plots uniform colouring is followed by giving the colours brown for pressure, +navy blue for energy density, sky blue for EoS parameter, blue for SEC, green for NEC and +red for DEC. Figures 4,5 and 6 illustrate the variation of pressure against redshift in cases I, II +and III respectively. The figures shows that in three cases pressure is negative and it is known +that a negative pressure fluid is the correct mechanism which is capable of explaining cosmic +acceleration within the standard cosmologies, despite the fact that in the latter it is necessary +to bring the cosmological constant to get this exotic characteristic. In pressure graphs increase +with the decrease of redshift that it is increases as the time increases is perceived which repre- +sents cosmic acceleration. +Figure 4: Pressure(p) +in case I +Figure 5: Pressure(p) +in case II +Figure 6: Pressure(p) +in case III +vi) Figures 7,8 and 9 shows the evolution of energy density for cases I, II and III respectively. In +all the cases the density decreases with the decrease of redshift i.e. decreases as time increases. +vii) With great efforts the equation of state(EoS) parameter in cosmology of different dark +Figure 7: Energy density(ρ) +in case I +Figure 8: Energy density(ρ) +in case II +Figure 9: Energy density(ρ) +in case III +energy models are examined. +The parameter relating to the equation of state is a dimen- +sionless term that represents the matter state under some particular physical grounds. In the +terminology of p and ρ the EoS can be interpret in the from of ω = p +ρ. The EoS parameter +11 + +-0.06 +-0.08 +pressure(p) +-0.1 +-0.12 +-0.14 +-0.16 +-0.18 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)-0.08 +-0.1 +pressure(p) +-0.12 +-0.14 +-0.16 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)-0.04 +-0.06 +pressure(p) +-0.08 +-0.1 +-0.12 +-0.14 +-0.16 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)0.2 +0.18 +energy density(p) +0.16 +0.14 +0.12 +0.1 +0.08 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)0.22 +0.2 +0.18 +0.16 +0.12 +0.1 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)0.18 +0.16 +energy density(p) +0.14 +0.12 +0.1 +0.08 +0.06 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)is distinguished in three regions namely quintessence, phantom, and quintom according to its +range. In quintessence region the EoS paramter lies in the range of −1 < ω < − 1 +3, in phantom +phase the EoS parameter is in the range of less than -1 (i.e. ω < −1) and in quintom ω = −1. +Figures 10, 11 and 12 of EoS parameter are drawn against redshift and observe that decrease +with the decrease of redshift that is decrease as time increases. From the graphs we noticed that +our model lies in quintessence region in three cases. According to Planck+nine years WMAP +the current value of EoS parameter is approximately as ω = −1.13+0.24 +−0.25 [58], and from SNe Ia +data with galaxy clustering, CMBR anisotropy statistics the EoS parameter lies in the range +−1.33 < ω < −0.79, −1.67 < ω < −0.62 [59] respectively. From the figures of EoS parameters, +it is seen that three cases are approximately coincide with observational data which is a good +result. +viii)In modified theories of gravity, energy conditions [60–62] plays a crucial role in studying +Figure 10: EoS parameter(ω) +in case I +Figure 11: EoS parameter(ω) +in case II +Figure 12: EoS parameter(ω) +in case III +the behaviour of spacelike and timelike geodesics and these conditions are came from Ray- +chaudhuri equations [63]. Energy conditions can be defined in many ways, such as geometric +way and physical way. Moreover energy conditions are significant in the black hole physics, +as they lay foundations of the singularity theorems. Another advantage of energy condition is +that it allows basic tools to consider certain ideas about black holes and wormholes. There are +four most commonly used fundamental energy conditions. The general expressions for energy +conditions in regard of pressure and energy density are given below: +(i)SEC(Strong Energy condition): +Gravity always has to be attractive, and in cosmology +ρ + 3p ≥ 0, ρ + p ≥ 0 should be observed. +(ii)DEC(Dominant Energy Condition): The energy density should always be positive when +measured by any observer that is ρ ≥ 0, ρ ± p ≥ 0, must be obeyed. +(iii)WEC(Weak Energy Condition): The energy density must always be positive when mea- +sured by any observer that is ρ ≥ 0, ρ + p ≥ 0. +(iv)NEC(Null Energy Condition): NEC is expressed in the form of ρ+p ≥ 0 and it ensures the +validity of second law of black hole thermodynamics. +12 + +-0.78 +-0.79 +-0.8 +0.81 +-0.82 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)-0.75 +0.76 +-0.77 +0.78 +0.79 +-0.8 +-0.81 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)-0.66 +0.68 +-0.7 +0.72 +0.74 +-0.76 +-0.78 +-0.5 +0 +0.5 +1 +1.5 +2 +2.5 +redshift(z)Where NEC, WEC, DEC and SEC represents null, weak, dominant and strong energy condi- +tions. According to present cosmological data to represent the universe with cosmic expansion +the SEC of that model should be violated (ρ + 3p ≤ 0). For the obtained models the same +scenario can be clearly observed from figures 13 to 15. When compared to strong energy condi- +tion null energy condition is more beneficial, as it can be used algebraically due to its weakest +pointwise energy condition which results in the strongest theorems and all these energy condi- +tions, are met by electromagnetic field. From figures 16 to 18 it is clear that NEC(ρ + p ≥ 0) +is satisfied in all the three cases for the obtained model. If NEC satisfies then the parameter +EoS occurs in quintessence region. Also from figures 19 to 21 it is clear that DEC (ρ + p ≥ 0) +is fulfilled in all the three cases for the obtained model. +Figure 13: SEC in case I +Figure 14: SEC in case II +Figure 15: SEC in case III +Figure 16: NEC in case I +Figure 17: NEC in case II +Figure 18: NEC in case III +Figure 19: DEC in case I +Figure 20: DEC in case II +Figure 21: DEC in case III +13 + +-0.1 +-0.15 +SEC(p+3p) +-0.2 +-0.25 +-0.3 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)-0.14 +0.16 +SEC(p+3p) +-0.18 +0.2 +-0.22 +-0.24 +-0.26 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)-0.05 +-0.1 +SEC(p+3p) +-0.15 +-0.2 +-0.25 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)0.045 +0.04 +NEC(p+p) +0.035 +0.03 +0.025 +0.02 +0.015 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)0.06 +0.05 +NEC(p+p) +0.04 +0.03 +0.02 +0.01 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)0.05 +0.03 +0.02 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)0.35 +0.3 +DEC(p-p) +0.25 +0.2 +0.15 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)0.4 +0.35 +DEC(p-p) +0.3 +0.25 +0.2 +0.15 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)0.3 +0.2 +0.15 +0.1 +-0.5 +0 +0.5 +1 +1.5 +2 +redshift(z)4.1 +Stability analysis +Perturbations are essential for simplify a complex mathematical problems. There are several +types of perturbations such as isotropic, anisotropic, homogeneous/inhomogeneous scalar, vec- +tor and tensor perturbations. The technique of perturbation is studied as a tool for finding +approximate solution and comparing it to the obtained exact solution. Some of the researchers +who studied on stability analysis are [64–66]. Here the stability of solutions in terms of metric +perturbation as following +ai → aBi + δai = aBi(1 + δbi). +(42) +The perturbation of volume scale factor, directional Hubble factors and mean Hubble factors +are +V → VB + VB +� +i +δbi, +θi → θBi + +� +i +δbi, +θ → θB + 1 +3 +� +i +δbi. +(43) +The following equations are satisfied by the metric perturbation δbi +� +i +δ¨bi + 2 +� +i +θBiδ˙bi = 0, +(44) +δ¨bi + +˙VB +VB +δ˙bi + +� +j +δ˙bjθBi = 0, +(45) +� +i +δ˙bi = 0. +(46) +From equations (44) - (46), we attain +δ¨bi + +˙VB +VB +δ˙bi = 0, +(47) +where VB is the background spatial volume and for our case VB is +VB = (sinh(αt)) +3 +β . +(48) +From above two equations, δbi is procured as +δbi = c1 − c +�β +� +cosh2(αt) sech(αt) sinh +β−3 +β (αt)2F1 +� +1 +2, β−3 +2β ; 3(β−1) +2β +; − sinh2(αt) +� +α(β − 3) +� +, +(49) +where c1 and c are integrating constants. +consequently, the actual fluctuations δai = aBiδbi is +δai = +� +c1−c +�β +� +cosh2(αt) sech(αt) sinh +β−3 +β (αt)2F1 +� +1 +2, β−3 +2β ; 3(β−1) +2β +; − sinh2(αt) +� +α(β − 3) +�� +(sinh(αt)) +−3 +β . +(50) +14 + +Figure 22: Plot of actual fluctuations(δai) versus redshift(z) +Figure 22 shows the behaviour of actual fluctuations versus redshift and it is noticed that it +is a decreasing function with the decrease of redshift that is decreases as time increases. It is +clear that δai → 0 as z → −∞ and hence the background solution is shown to be stable against +perturbation of gravitational field. +5 +Conclusions +A cosmological model in f(R, T) theory with three cases namely power law, logarithmic and +exponential curvature is obtained. Hyperbolic scale factor is used to solve the field equations to +get the solution in each case. The solutions of these field equations represent accelerating model +of the universe. The graph of all parameters are drawn against redshift. In graphs the negative +region of z represents future epoch, positive region of z represents past and z = 0 indicates +present. Obtained models are anisotropic and free from singularity all the way through the +universe’s evolution. By analyzing all the parameters the conclusions are as follows: +• From figures 1 and 3 and from the equations (34) and (36) it can be seen that Hubble +parameter and expansion scalar decreases with the decrease of redshift, and also it is clear +that the Hubble parameter and expansion scalar are close to zero when t → ∞. +• From figure 2 it is clear that volume increases with the decrease of redshift which indicates +volume of the expanding universe. From equation (37), it is noticed that the shear scalar +is a function of time and tends to zero when t → ∞. +• From equation (39), the anisotropic parameter is independent of time and Ah ̸= 0 for +n ̸= 1, Ah = 0 for n = 1. But in this paper due to power law n is different from one. +Hence the models are anisotropic throughout. +15 + +0.08 +0.06 +0.04 +0.02 +0 +-1 +0 +1 +2 +3 +redshift(z)• From the graphs of pressure and energy density of all the three cases, it is clear that the +pressure and energy density are negative and positive respectively. Due to the negative +pressure and positive energy density the universe is going through accelerating expansion. +• The behavior of EoS parameter against redshift is represented in plots 11 to 13. From +these graphs it is obvious that the model is in the quintessence region in three cases that +is −1 < ω < − 1 +3 which matches with present observational data. +• In three cases, SEC is violated whereas NEC and DEC are fufilled. The violation of SEC +leads to cosmic acceleration which is in good agreement with the expansion of the cosmos. +• As seen in the graph of stability analysis, the actual fluctuations begin with a small +positive value and decreases to zero. As a result, the background solution is stable when +the gravitational field is perturbed. +A detailed discussion is provided through the obtained models for describing cosmic accel- +eration. Finally, through the detailed study of the models in three cases namely power law +curvature f(R, T) = R+γR2 − µ4 +R +λT, logarithmic curvaturef(R, T) = R+νln(τR)+λT and +exponential curvature f(R, T) = R+κe−ιR+λT very good results which represents the universe +accelerating expansion are observed. Moreover all the parameters discussed here matches with +the recent observational data. At last, without existence of any exotic fluid, the current uni- +verse is accelerating is perceived in this paper which is a great outcome. As a future work, this +work can be extended to other anisotropic models and can study the similarities and differences +between them. +6 +Appendix +The values of χ, ξ, η are same for all the three cases and are given below +χ = ϱ1 +ϱ2 +where +ϱ1 = β2 sinh(αt)2(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2 sinh(αt) +−6+(−2n−4)β +β(n+2) +−6(−9 cosh(αt)2 +2 ++ β(n + 2))α2 +ϱ2 = β2(n + 2)2 sinh(αt)2 +ξ = ϱ3 +ϱ2 +16 + +where +ϱ3 = −3 +� +(−3n2 − 3n − 3) cosh(αt)2 + β(n + 2)(n + 1) +� +α2 +η = ϱ4 +ϱ2 +ϱ4 = β2 sinh(αt)2(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2 sinh(αt) +−6+(−2n−4)β +β(n+2) ++18α2(n + 1 +2) cosh(αt)2 +For case(i) The values of φ1, φ2, φ3, φ4, φ5, φ6 and φ7 are given below +φ1(t) = ϱ5 +ϱ6 +where +ϱ5(t) = 288(n + 2)2 sinh(αt) +6+(2n+4)β +β(n+2) β2� +− +� +(n + 2)2β − 3n2 − 6n − 9 +� +α2 cosh(αt)2 sinh(αt) +6 +β(n+2) ++ +� +α2 sinh(αt) +6+(2n+4)β +β(n+2) ++ cosh(αt)2β +3 +− β +3 +� +(n + 2)2β +�2 +(π + 3λ +16) +ϱ6(t) = ϱ7(t) + 4(cosh(αt) − 1)(n + 2)2β2(cosh(αt) + 1)ϱ8(t) +ϱ7(t) = +�� +µ4(n + 2)6β6 + 324α4(n + 2)2(n2 + 2n + 3)2β2 − 11664α6(n2 + 2n + 3)3γ +� +cosh(αt)6 − 3(n + 2)2β +� +µ4(n + 2)4β5 + 72α4(n2 + 2n + 3)(n + 2)2β2 + 108α4 +(n2 + 2n + 3)2β − 3888α6(n2 + 2n + 3)2γ +� +cosh(αt)4 + 3(n + 2)4β2� +µ4(n + 2)2β4 ++ 12α4(n + 2)2β2 + 72α4(n2 + 2n + 3)β − 1296α6(n2 + 2n + 3)γ +� +cosh(αt)2 +− β3(n + 2)6(−432α6γ + µ4β3 + 36α4β) +� +sinh(αt) +18 +β(n+2) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +ϱ8(t) = −6 +� +(−3n2 − 6n − 9) cosh(αt)2 + (n + 2)2β +� +α2� +(−54α2(n2 + 2n + 3)γ ++ β2(n + 2)2) cosh(αt)2 − β(n + 2)2(−18α2γ + β) +� +sinh(αt) +12 +β(n+2) + (n + 2)2β2 +(cosh(αt) − 1)(cosh(αt) + 1) +� +((−108α2(n2 + 2n + 3)γ + β2(n + 2)2) cosh(αt)2 +− β(n + 2)2(−36α2γ + β)) sinh(αt) +6 +β(n+2) − 4β2γ(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +φ2(t) = ϱ9(t) +ϱ6(t) +where +ϱ9(t) = λ18(n + 2)2 sinh(αt) +6+(2n+4)β +β(n+2) β2� +− +� +(n + 2)2β − 3n2 − 6n − 9 +� +α2 cosh(αt)2 +sinh(αt) +6 +β(n+2) + (n + 2)2β +� +α2 sinh(αt) +6+(2n+4)β +β(n+2) ++ β sinh(αt)2 +3 +��2 +17 + +φ3(t) = ϱ10 +ϱ13 +where +ϱ10(t) = −2 +� +ϱ11 − ϱ12 + β2(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2� +ϱ11(t) = +�� +µ4(n + 2)6β6 − 2916α6γ(n2 + 2n + 3)3� +cosh(αt)6) − 3β(n + 2)2� +µ4(n + 2)4β5 +− 972α6γ(n2 + n + 3)2� +cosh(αt)4 + 3 +� +µ4(n + 2)4β4 − 324α6γ(n2 + 2n + 3) +� +β2 +(n + 2)4 cosh(αt)2 − β3(n + 2)6(−108α6γ + β3µ4) +� +sinh(αt) +18 +β(n+2) +� +� +� +� +� +� +� +� +� +� +� +ϱ12(t) = 108β2(n + 2)2γ(cosh(αt) + 1) +� +(−3n2 − 6n − 9) cosh(αt)2 + (n + 2)2β +�2 +α4 +(cosh(αt) − 1) sinh(αt) +12 +β(n+2) + 36β4(n + 2)4γ(cosh(αt) + 1)2� +(−3n2 − 6n − 9) +cosh(αt)2 + (n + 2)2β +� +α2(cosh(αt) − 1)2 sinh(αt) +6 +β(n+2) − 4γβ6(cosh(αt) − 1)3 +(cosh(αt) + 1)3(n + 2)6� +(−3n2 − 6n − 9) cosh(αt)2 + (n + 2)2β +� +α2 sinh(αt) +6 +β(n+2) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +ϱ13(t) = (ϱ7 − ϱ14) sinh(αt) +6 +β(n+2)(n + 2)2β2 sinh(αt)2 +ϱ14(t) = −24 +�� +(n + 2)2β2 − 54α2γ(n2 + 2n + 3) +� +cosh(αt)2 − β(n + 2)2(−18α2γ + β) +� +(n + 2)2α2(cosh(αt) − 1)β2(cosh(αt) + 1)((−3n2 − 6n − 9) cosh(αt)2 ++ (n + 2)2β) sinh(αt) +12 +β(n+2) + 4(n + 2)4�� +(n + 2)2β2 − 108α2γ(n2 + 2n + 3) +� +cosh(αt)2 − β(n + 2)2(−36α2γ + β) +� +(cosh(αt) − 1)2β4(cosh(αt) + 1)2 +sinh(αt) +6 +β(n+2) − 16γβ6(cosh(αt) − 1)3(cosh(αt) + 1)3(n + 2)6 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +φ4(t) = −36α2ϱ15ϱ16 +(ϱ7 − ϱ14)ϱ17 +where +ϱ15(t) = +� +µ4(n + 2)6β6 + 5832α6γ(n2 + 2n + 3)3� +cosh(αt)6 − 3β(n + 2)2� +µ4(n + 2)4β5 ++ 1944α6γ(n2 + 2n + 3)2� +cosh(αt)4 + 3β2(n + 2)4� +µ4(n + 2)2β4 + 648α6γ +(n2 + 2n + 3) +� +cosh(αt)2 − β3(n + 2)6(216α6γ + β3µ4) +� +sinh(αt) +18 +β(n+2) + ϱ25 +� +� +� +� +� +� +� +� +� +� +� +ϱ25(t) = 216(cosh(αt) + 1)β2(cosh(αt) − 1)(n + 2)2γα4� +(−3n2 − 6n − 9) cosh(αt)2 ++ β(n + 2)2�2 +sinh(αt) +12 +β(n+2) − 72(cosh(αt) + 1)2β4(cosh(αt) − 1)2(n + 2)4γα4 +� +(−3n2 − 6n − 9) cosh(αt)2 + β(n + 2)2� +sinh(αt) +6 +β(n+2) + 8γβ6(cosh(αt) − 1)3 +(cosh(αt) + 1)3(n + 2)6 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +18 + +ϱ16(t) = +� +(β(n + 2)2 − 3n2 − 6n − 9) sinh(αt) +6 +β(n+2) − β(cosh(αt) − 1)(cosh(αt) + 1) +(n + 2) +� +cosh(αt)2 +ϱ17(t) = (−3α2((−3n2 − 6n − 9) cosh(αt)2 + (n + 2)2β) sinh(αt) +6 +β(n+2) + β2(cosh(αt) − 1) +(cosh(αt) + 1)(n + 2)2)β(n + 2) sinh(αt)2 +φ5(t) = +24ϱ15ϱ18 +(ϱ7 − ϱ14)ϱ19 +where +ϱ18(t) = +� +((n + 2)2β − 3n2 − 6n − 9)α2� +cosh(αt)2 + 1 +2 +� +sinh(αt) +6 +β(n+2) +−(cosh(αt) − 1)(6 cosh(αt)2 + β(n + 2))(cosh(αt) + 1) +2 +� +α2 +ϱ19(t) = −3α2((−3n2 − 6n − 9) cosh(αt)2 + (n + 2)2β) sinh(αt) +6 +β(n+2) ++β2(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2 +φ6(t) = +ϱ20 +ϱ21ϱ22 +where +ϱ20(t) = − +� +((n + 2)2β − n2 − 6n − 9)α2 sinh(αt) +6+(2n+4)β +β(n+2) +− ((n + 2)2) − 3n2 − 6n − 9) cosh(αt)2 +α2 sinh(αt) +6 +β(n+2) + β sinh(αt)2(n + 2) +�2 +sinh(αt) +18+(4n+8)β +β(n+2) +β6 cosh(αt)2(n + 2)6α2µ4 +ϱ21(t) = +� +α2β(n + 2)2 sinh(αt) +6+(2n+4β) +β(n+2) +− ((n + 2)2β − 3n2 − 6n − 9) cosh(αt)2α2 sinh(αt) +6 +β(n+2) ++β2 sinh(αt)2(n + 2)2 +3 +�2 +ϱ22(t) = ϱ23 + ϱ24(n + 2)2β2(cosh(αt) − 1)(cosh(αt) + 1) +ϱ23(t) = +� +ϱ26 + 1 +8((n + 2)2β(µ4(n + 2)4β5 + 72α4(n2 + 2n + 3)(n + 2)2β2 ++ 108α4(n2 + 2n + 3)2β − 3888α6(n2 + 2n + 3)2γ) cosh(αt)4) − 1 +8(n + 2)4β2 +(µ4(n + 2)2β4 + 12α4(n + 2)2β2 + 72α4(n2 + 2n + 3)β − 1296α6(n2 + 2n + 3) +γ) cosh(αt)2 + 1 +24β3(n + 2)6(432α6γ + µ4β3 + 36α4β) +� +sinh(αt) +18 +β(n+2) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +ϱ26(t) = +�−µ4(n + 2)6β6 +24 +− 27α4(n + 2)2(n2 + 2n + 3)2β2 +2 ++ 486α6(n2 + 2n + 3)3γ +� +cosh(αt)6 +19 + +ϱ24(t) = +� +((−3n2 − 6n − 9) cosh(αt)2 + (n + 2)2β)α2((−54α2(n2 + 2n + 3)γ + β2(n + 2)2) +cosh(αt)2 − β(n + 2)2(−18α2γ + β)) sinh(αt) +12 +β(n+2) + 1 +3 +� +2(n + 2)2β2�� +(−β2(n + 2)2 +4 ++ 27α2(n2 + 2n + 3)γ) cosh(αt)2 + 1 +4β(n + 2)2(−36α2γ + β) +� +sinh(αt) +6 +β(n+2) + β2γ +(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2� +(cosh(αt) − 1)(cosh(αt) + 1) +�� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +φ7(t) = −36nα2ϱ15ϱ16 +ϱ17(ϱ7 − ϱ14) +For case(ii) The values of φ1, φ2, φ3, φ4, φ5, φ6 and φ7 are given below +φ1(t) = 48δ3(π + 3λ +16) +δ1 +where +δ1 = +��� +ν(n + 2)2β2 − 18α2(n2 + 2n + 3) +� +cosh(αt)2 − β(n + 2)2(−6α2 + βν) +� +sinh(αt) +6 +β(n+2) − 2β2(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2� +δ3 = +�� +(−3n2 − 6n − 9) cosh(αt)2 + β(n + 2)2� +α2 sinh(αt) +6 +β(n+2) +−β2 sinh(αt)2(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2 +3 +� +φ2(t) = −δ2λ +δ1 +where +δ2 = −3 +� +(−3n2 − 6n − 9) cosh(αt)2 + β(n + 2)2� +α2 sinh(αt) +6 +β(n+2) ++β2(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2 +φ3(t) = −3νδ3δ4 +δ1 +where +δ4 = ln +� +1 +β2(n + 2)2 sinh(αt)2 +� +δ9 +� ++ ln(2) + ln(τ) − 1 +� +20 + +δ9 = −β2(cosh(αt) − 1)2(cosh(αt) + 1)2(n + 2)2 sinh(αt) +−6+(−2n−4)β +β(n+2) ++3α2� +(−3n2 − 6n − 9) cosh(αt)2 + β(n + 2)2� +φ4(t) = δ5 +δ2δ1 +where +δ5 = 18 +� +− β(cosh(αt) − 1)(cosh(αt) + 1)(n + 2) sinh(αt) +6 +β(n+2) + (β(n + 2)2 +−3n2 − 6n − 9) sinh(αt) +12 +β(n+2)α2� +(n + 2)νβ cosh(αt)2α2 +φ5(t) = δ6 +δ2δ1 +where +δ6 = −12(n + 2)2νβ2� +δ10 + (β(n + 2)2 − 3n2 − 6n − 9)(cosh(αt)2 + 1 +2) sinh(αt) +12 +β(n+2)α2� +α2 +δ10 = −(cosh(αt) − 1)(cosh(αt) + 1)(6 cosh(αt)2 + β(n + 2)) sinh(αt) +6 +β(n+2) +2 +φ6(t) = δ7 +δ8δ1 +where +δ7 = 4(n + 2)2 sinh(αt) +6 +β(n+2)νβ2 cosh(αt)2� +(β(n + 2)2 − 3n2 − 6n − 9)α2] sinh(αt) +6 +β(n+2) +−β(cosh(αt) − 1)(cosh(αt) + 1)(n + 2) +�2 +α2 +δ8 = +� +− (β(n + 2)2 − 3n2 − 6n − 9) cosh(αt)2α2 sinh(αt) +6 +β(n+2) + δ11 +�2 +δ11 = (n + 2)2β +� +α2 sinh(αt) +6+(2n+4)β +β(n+2) ++ β sinh(αt)2 +3 +� +φ7(t) = nδ5 +δ2δ1 +For case(iii) The values of φ1, φ2, φ3, φ4, φ5, φ6 and φ7 are given below +φ1(t) = −16π − 3λ +2ζ1 − 2 +21 + +where +ζ1 = κιe +−2ι +� +ζ8−3α2 +� +(−3n2−6n−9) cosh(αt)2+β(n+2)2 +�� +β2(n+2)2 sinh(αt)2 +ζ8 = β2(cosh(αt) − 1)2(cosh(αt) + 1)2(n + 2)2 sinh(αt) +−6+(−2n−4)β +β(n+2) +φ2(t) = − +λ +2ζ1 − 2 +φ3(t) = ζ2ζ4 +ζ3 +where +ζ2 = e +2 +� +β2 sinh(αt)2(cosh(αt)4+1)(n+2)2 sinh(αt) +−6+(−4n−8)β +β(n+2) ++9α2 cosh(αt)2(n2+2n+3) +� +ι +β2(n+2)2 sinh(αt)2 +ζ3 = β2(n + 2)2 sinh(αt)4� +ικe +2ι +� +β2(cosh(αt)4+1)(n+2)2 sinh(αt) −6+(−2n−4)β +β(n+2) ++9α2 cosh(αt)2(n2+2n+3) +� +β2(n+2)2 sinh(αt)2 +−e +4ι +� +sinh(αt) −6+(−2n−4)β +β(n+2) +cosh(αt)2β+ 3α2 +2 +� +β sinh(αt)2 +� +ζ4 = κ +� +β2ι(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2 sinh(αt) +−6 +β(n+2) + +�−(n + 2)2β2 +2 ++9α2ι(n2 + 2n + 3) +� +cosh(αt)2 − 3(n + 2)2β(ια2 − β +6 ) +� +φ4(t) = −36α2ζ5ζ2ι2κ +β(n + 2)ζ3 +where +ζ5 = β(cosh(αt) − 1)(cosh(αt) + 1)(n + 2) sinh(αt) +−6 +β(n+2) + α2� +(n + 2)2 +β − 3n2 − 6n − 9 +� +cosh(αt)2 +φ5(t) = −24ι2ζ2α2κζ6 +ζ3 +where +ζ6 = −ζ8 + +� +cosh(αt)2 + 1 +2 +� +((n + 2)2β − 3n2 − 6n − 9)α2 +ζ8 = (cosh(αt) − 1)(6 cosh(αt)2 + β(n + 2))(cosh(αt) + 1) sinh(αt) +−6 +β(n+2) +2 +φ6(t) = −144α2ι2 cosh(αt)2ζ1ζ7 +(n + 2)4β4(ζ1 − 1) +φ7(t) = −36nα2ζ5ζ2ι2κ +β(n + 2)ζ3 +ζ7 = +� +α2((n + 2)2β − 3n2 − 6n − 9) sinh(αt) +6 +β(n+2) − β(cosh(αt) − 1) +(cosh(αt) + 1)(n + 2) +�2 +sinh(αt) +−12+(−6n−12)β +β(n+2) +22 + +References +[1] A. 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Space Sci. 342 (2012) 257 [arXiv:1108.2133] +27 + diff --git a/n9AzT4oBgHgl3EQfOPuo/content/tmp_files/load_file.txt b/n9AzT4oBgHgl3EQfOPuo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d28e0f7516271bd3c03bf27ead53befd2aaed34a --- /dev/null +++ b/n9AzT4oBgHgl3EQfOPuo/content/tmp_files/load_file.txt @@ -0,0 +1,1562 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf,len=1561 +page_content='The study of Kantowski-Sachs perfect fluid cosmological model in modified gravity 1T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Vinutha, 1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Niharika and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Sri Kavya2 1Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' of Applied Mathematics, AUCST, Andhra University, Visakhapatnam, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' 2 Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' of Mathematics, Maharaj Vijayaram Gajapathi Raj College of Engineering, Vizianagaram-535005, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' ⋆vinuthatummala@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='com Abstract Kantowski-Sachs perfect fluid cosmological model is explored in modified gravity with functional form f(R, T)=f1(R)+f2(T) where R is Ricci scalar and T is the trace of energy- momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' With this functional form, three different cases have been formulated, namely negative and positive powers of curvature, logarithmic curvature and exponential curvature given by f1(R) = R + γR2 − µ4 R , f1(R) = R + νln(τR) and f1(R) = R + κe−ιR respectively, and for all these three cases, f2(T) = λT, here γ, λ, µ, ν, τ, κ and ι are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' While solving the field equations, two constraints i) Expansion scalar is proportional to shear scalar ii) Hyperbolic scale factor are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' By using these conditions the required optimum solutions are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The physical parameters are calculated and geometrical parameters of three cases are analysed against redshift(z) with the help of pictorial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In the context of f(R, T) gravity energy conditions are discussed with the help of pressure and energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' If strong energy condition is positive the gravity should be attractive but in our model it shows negative it means that cosmic acceleration is due to antigravity, whereas NEC and DEC are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The perturbation technique is used to test the stability of the background solutions of the obtained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The inferences obtained from this paper are in persistent with the present cosmological observations and the model represents an accelerating universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Keywords: Kantowski-Sachs spacetime, f(R, T) theory, perfect fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' 1 Introduction Einstein’s theory of general relativity is the foundation of modern physics and it describes black holes and gravitational phenomena but it break down to give an explanation of cosmic acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In recent scenario it is well known that our universe is accelerating [1, 2] and it is one of the trending topics in cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' To understand this mysterious concept, we focused on dark energy and modified theories of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The universe is going through an accelerated 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='01163v1 [gr-qc] 3 Jan 2023 period of expansion and it is revealed by the experiments such as CMBR and SNIa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Dark energy can be inspected in many ways and reforming the geometric part of the Einstein-Hilbert action is regarded as the most efficient possible way and these changes lead to so many alter- native theories of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' There are different classes of modified gravity such as f(R) gravity, f(T) gravity, f(G) gravity, f(R, G) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Among them f(R) gravity has attracted many researchers because it provides a natural gravitation alternative to dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' During the universe expansion f(R) theory elucidate the change from deceleration phase to acceleration phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' f(R) theory is presumed to be beneficial for resolution of the hierarchy problem or uni- fication of grand unified theories with gravity in high energy physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Sotiriou and Faraoni [3] have worked on f(R) theories of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Antonio De Felice [4] and Shinji Tsujikawa have studied on f(R) theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' A new class of f(R, T) gravity presented by Harko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' [5], by including trace T in f(R) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The T-dependence in f(R, T) gravity may appear from the presence of imperfect fluids or quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Among all the modified theories of gravitation, the f(R, T) theory is a generalized theory because there is an energy transfer relation between matter and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The existence of this relationship is the cause of the rapid expansion of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The authors who worked on f(R, T) gravity are included in references [6–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In this paper, we examine three specific cases one of them is combination of 1 Rx and Ry i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' f(R) = R+γRy − µ4 Rx where γ and µ are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In this functional form, it has both positive and negative curvature powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' At low curvature it leads to gravitational alternative for dark energy which helps in speed up of cosmic expansion where as high curvature describes the infla- tionary stage of early universe [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' By considering Ry term for 1 < y < 2 power law inflation happen at early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' If y = 2, Starobinsky inflation takes place [15], the term R2 indicates nat- ural correction to general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' According to Nojiri and Odinstov [16] R2 term is necessary to get rid of instabilities, linear growth of the gravitational force, produce early time inflation and appear to pass the solar system tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The state of no linear growth for gravitational force makes it very much fascinating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Higher derivative terms like R2, R3 can be used to put down the instabilities significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' For equivalent scalar tensor theory the solar system test may be passed as scalar has large mass originated again by higher derivative terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The standard Einstein’s gravity may be modified by considering a 1 R term in the Einstein Hilbert action [17] which represents the present acceleration of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' But the insertion of 1 R term gener- ates instabilities which can be overcome by addition of R2 term to the Einstein’s gravitational action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Besides the advantages of this functional form, have well acceptable Newtonian limit, no instabilities and no Brans Dicke problem in scalar tensor version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' When we put y = 2 and x = 1 in the above functional form f(R) = R+γRy − µ4 Rx it reduces to f(R) = R+γR2− µ4 R and the obtained results are very efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In addition to this functional form by using the linear function of f(T) = λT, we get the final form of f(R, T) = R + γR2 − µ4 R + λT where γ, µ and λ 2 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Vinutha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' [18] have worked on Kantowski–Sachs perfect fluid cosmological model in R2- Gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Vinutha and Sri Kavya [19] have studied Bianchi type cosmological mod- els in f(R, T) theory with quadratic functional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Brookfield [20] have worked on viability of f(R) theories with additional powers of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Godani and Samanta [21] have studied traversable warmholes on f(R) gravity where f(R) = R+αRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Banik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' [22] have discussed Bianchi-I cosmological model in f(R) = R − β Rn gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Next, we consider logarithmic curvature i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' f(R, T) = R + νln(τR) + λT where τ, ν and λ are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' As this modified gravity has put forward a gravitational alternative for dark en- ergy, it is quite interesting to work on this particular functional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In this model logarithmic terms are produced by quantam effects in curved space time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The need for dark energy may be eradicated by this modified gravity and may aid for the fusion of the early time inflation and cosmic acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Nojiri and Odinstov have studied about modified gravity and proposed some functional forms such as ln(R) or R−n(lnR)m and R + γR−n(ln R µ2)m [23,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Fayyaz and Shamir [25] have analysed wormhole structures in logarithmic-corrected R2 gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Kourosh and Tahereh [26] have discussed phantom-like behavior in f(R) = R + βlog( R µ2) + γRm gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' By appending the torsion scalar component to the exponential f(R) theory [27–31], the functional form is f(R, T) = R + κe−ιR + λT where κ, ι and λ are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The reason behind choosing this functional form it comes up with the best way of exploring cosmic acceler- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In contrast to the ΛCDM model the exponential gravity model has one more parameter included in it and it also permits the relaxation of fine tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Vinutha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' [32] have stud- ied on Bianchi type cosmological models in modified theory with exponential functional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Paul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' [33] have worked on accelerating universe in modified theories of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' [34] have studied on f(R, T) = f(R) + λT gravity models as alternatives to cosmic ac- celeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Moreas and Sahoo [35] have discussed traversable wormholes by using functional form f(R, T) = R + γeχT and also with this functional form Moreas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' [36] studied FRW cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' When compared to other anisotropic metrics, Kantowski-Sachs model is very simple and easy to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The cosmologies of Kantowski-Sachs metric possess two properties of symme- try such as spherical symmetry and invariance under spatial translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' It describes spatially homogeneous, anisotropic universe and interior of black holes that does not allow a simply transitive group of motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' It is also used to analyze the behavior of the added degrees of freedom in quantum cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' This metric represents three different anisotropic 3 + 1 dimensional space time and positive curvature models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The study of anisotropic models were nourished by the theoretical studies and observations of CMB which also been extended to modified theories of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Thus this model with an anisotropic nature appeared most appropriate in describing the early stage of the cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' some of the authors who worked on 3 Kantowski Sachs model are [37–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' This article is organized as follows: In section 2, f(R, T) gravity field equations are obtained and in section 3 the field equations of power-law, logarithmic and exponential functional forms are solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Section 4 discusses the physical and geometrical properties of three cases using graphs and section 5 concludes our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' 2 A brief review of f(R, T) = f1(R) + f2(T) model The final action principle of f(R, T) gravity which is a function of matter Lagrangian Lm is read as S = � � 1 16πf(R, T) + Lm � √−gd4x, (1) where g is the metric determinant of the fundamental tensor gij, f(R, T) is an arbitrary function of R and T which is mentioned in the abstract, Lm is the usual matter Lagrangian density and we consider G = c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' By varying the above equation (1) with respect to gij,we obtain the field equations of f(R, T) gravity in covariant tensor form as fR(R, T)Rij − 1 2f(R, T)gij + (gij□ − ∇i∇j)fR(R, T) = 8πTij − fT(R, T)θij − fT(R, T)Tij, (2) here, ∇i is the covariant derivative and □ = ∇i∇j is the D’Alemberts operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' fR = ∂f(R,T) ∂R , fT = ∂f(R,T) ∂T and Rij is the Ricci tensor, where θij = −2Tij + gijLm − 2glk ∂2Lm ∂gijglk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (3) Here the energy-momentum tensor is considered to be a perfect fluid which is defined as Tij = (p + ρ)uiuj − pgij, (4) where ui denotes four velocity vector in co-moving coordinates i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' ui = (1, 0, 0, 0) and uiuj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Hence, the components of energy-momentum tensor become Tij=diag(ρ, −p, −p, −p), where p is the pressure and ρ is the energy density of perfect fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Several authors have studied by choosing energy-momentum tensor as perfect fluid which are included in the references [44–51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' It takes the form by replacing matter Lagrangian as Lm = −p [52–54] in equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' θij = −2Tij − pgij (5) 4 Consequently the field equations for f(R, T) gravity are procured with the aid of T = ρ − 3p in equation (2) as Gij = 1 fR(R, T) � [8π + fT(R, T)]Tij + pfT(R, T)gij + 1 2[f(R, T) − RfR(R, T)]gij − (gij□ − ∇i∇j)fR(R, T) � , (6) where Gij is the Einstein tensor which is expressed as Rij − 1 2Rgij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Here, we consider the functional form f(R, T) = f1(R) + f2(T) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' f(R, T) = R + γR2 − µ4 R + λT (7) f(R, T) = R + νln(τR) + λT (8) f(R, T) = R + κe−ιR + λT (9) as case I, II and III respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' 3 Metric and solutions of the field equations Now the metric takes the form, ds2 = dt2 − M 2(t)dr2 − N 2(t)(dθ2 + sin2 θdψ2, ) (10) where M and N are metric potentials and functions of cosmic time t only and co-moving coor- dinates are (r, θ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='1 Case I - (negative and positive powers of curvature) The functional form f(R, T) = R + γR2 − µ4 R + λT field equations are as follows 1 N 2 + 2 ¨N N + ˙N 2 N 2 = − (8π + 3λ 2 )p 1 + 2Rγ + µ4 R2 + λρ 2(1 + 2Rγ + µ4 R2) − ( γR2 2 + µ4 R ) 1 + 2Rγ + µ4 R2 − (2γ − 2µ4 R3 ) 1 + 2Rγ + µ4 R2 �2 ˙N N ˙R + ¨R � − 6µ4 ˙R2 R4 1 + 2Rγ + µ4 R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' � � � � � � � � � � � (11) ¨ M M + ¨N N + ˙M ˙N MN = − (8π + 3λ 2 )p 1 + 2Rγ + µ4 R2 + λρ 2(1 + 2Rγ + µ4 R2) − ( γR2 2 + µ4 R ) 1 + 2Rγ + µ4 R2 − (2γ − 2 µ4 R3) 1 + 2Rγ + µ4 R2 � ( ˙M M + ˙N N ) ˙R + ¨R � − 6 µ4 ˙R2 R4 1 + 2Rγ + µ4 R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' � � � � � � � � � � � (12) 5 2 ˙M ˙N MN + ˙N 2 N 2 + 1 N 2 = (8π + 3λ 2 )ρ 1 + 2Rγ + µ4 R2 − λp 2(1 + 2Rγ + µ4 R2) − ( γR2 2 + µ4 R ) 1 + 2Rγ + µ4 R2 − (2γ − 2 µ4 R3) 1 + 2Rγ + µ4 R2 � ˙M M + 2 ˙N N � ˙R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' � � � � � � � � � � � (13) here dot denotes derivate with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 Case - II (logarithmic curvature) Field equations corresponding to the f(R, T) = R + νln(τR) + λT are 1 N 2 + 2 ¨N N + ˙N 2 N 2 = −(8π + 3λ 2 )p 1 + ν R + λρ 2(1 + ν R) − ν(1 − ln(τR)) 2(1 + ν R) + �2 ˙N N ˙R + ¨R � ν R2 1 + ν R − 2ν ˙R2 R3 1 + ν R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' � � � � � � � � � (14) ¨ M M + ¨N N + ˙M ˙N MN = −(8π + 3λ 2 )p 1 + ν R + λρ 2(1 + ν R) − ν(1 − ln(τR)) 2(1 + ν R) + ν R2 1 + ν R � ( ˙M M + ˙N N ) ˙R + ¨R � − 2ν ˙R2 R3 1 + ν R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' � � � � � � � � � (15) 2 ˙M ˙N MN + ˙N 2 N 2 + 1 N 2 = (8π + 3λ 2 )ρ 1 + ν R − λp 2(1 + ν R) − ν(1 − ln(τR)) 2(1 + ν R) + ν R2 1 + ν R � ˙M M + 2 ˙N N � ˙R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' � � � � � � � � � (16) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='3 Case - III (exponential curvature) Field equations corresponding to the f(R, T) = R + κe−ιR + λT are given as follows: 1 N 2 + 2 ¨N N + ˙N 2 N 2 = −(8π + 3λ 2 )p 1 − κιe−ιR + λρ 2(1 − κιe−ιR) + κe−ιR(1 + ιR) 2(1 − κιe−ιR) − κι2e−ιR 1 − κιe−ιR � (2 ˙N N ) ˙R + ¨R � + κι3e−ιR ˙R2 1 − κιe−ιR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' � � � � � � � � � (17) ¨ M M + ¨N N + ˙M ˙N MN = −(8π + 3λ 2 )p 1 − κιe−ιR + λρ 2(1 − κιe−ιR) + κe−ιR(1 + ιR) 2(1 − κιe−ιR) − κι2e−ιR 1 − κιe−ιR � ( ˙M M + ˙N N ) ˙R + ¨R � + κι3e−ιR ˙R2 1 − κιe−ιR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' � � � � � � � � � (18) 6 2 ˙M ˙N MN + ˙N 2 N 2 + 1 N 2 = (8π + 3λ 2 )ρ 1 − κιe−ιR − λp 2(1 − κιe−ιR) + κe−ιR(1 + ιR) 2(1 − κιe−ιR) − κι2e−ιR ˙R 1 − κιe−ιR � ˙M M + 2 ˙N N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' � � � � � � � � � (19) To obtain solutions for highly non-linear equations is very strenuous and in order to remove such complications we require some constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (i) We consider σ is proportional to θ(where σ is the shear scalar and θ is the expansion scalar) and it generate linear relationship between two metric potentials in terms of M and N as M = N n, (20) n ̸= 0, 1 is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The physical motivation for assuming this condition is that Hubble expansion of the universe may attain isotropy by the observations of the velocity redshift relation for extra galatic sources if the value of σ θ is constant [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (ii) The average scale factor is assumed as a hyperbolic expansion a(t) = sinh(αt) 1 β , (21) where α > 0, β > 0 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The consequence of using this scale factor is time dependent deceleration parameter q [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' This average scale factor tends to zero if t → 0 and if t → ∞ then a(t) becomes infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The directional Hubble parameters are H1 = ˙M M H2 = H3 = ˙N N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (22) The average Hubble parameter is, H = 1 3(H1 + 2H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (23) By substituting equation (23) in equation (22), we get H = ˙a a = 1 3 � ˙M M + ˙N N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (24) From equations (20) - (24), we obtain metric potentials of M and N as M = (sinh(αt)) 3n β(n+2), (25) N = (sinh(αt)) 3 β(n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (26) If t → ∞ then M and N are nonzero, hence, our model is free from singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Using equations (25) and (26), the Kantowski sachs metric obtained as ds2 = dt2 − (sinh(αt)) 6n β(n+2) dr2 − (sinh(αt)) 6 β(n+2)(dθ2 + sin2 θdψ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (27) The above metric represents a perfect fluid Kantowski-Sachs universe in f(R, T) theory of grav- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='4 Pressure and energy density for case I By solving the equations of (11),(12) and (13) we get the pressure of the model as p = 1 4 �χ + ξ − 2η − φ4 + 2φ5 + 2φ6 − φ7 φ2 − φ1 − χ + ξ + 2η + 4φ3 + 7φ4 + 2φ5 + 2φ6 + 3φ7 φ1 + φ2 � , (28) and the energy density of the model is obtained as ρ = 1 4 �χ + ξ + 2η + 4φ3 + 7φ4 + 2φ5 + 2φ6 + 3φ7 φ1 + φ2 + χ + ξ − 2η − φ4 + 2φ5 + 2φ6 − φ7 φ2 − φ1 � , (29) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 Pressure and energy density for case II By solving the equations (14),(15) and (16) we get the expression for pressure is p = 1 4 �χ + ξ − 2η + φ4 − 2φ5 + 4φ6 + φ7 φ2 − φ1 − χ + ξ + 2η + 4φ3 − 7φ4 − 2φ5 + 4φ6 − 3φ7 φ1 + φ2 � , (30) and the energy density of the model is obtained as ρ = 1 4 �χ + ξ + 2η + 4φ3 − 7φ4 − 2φ5 + 4φ6 − 3φ7 φ1 + φ2 + χ + ξ − 2η + φ4 − 2φ5 + 4φ6 + φ7 φ2 − φ1 � , (31) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 Pressure and energy density for case III By solving the equations (17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (18) and(19) we get the pressure of the model as p = 1 4 �χ + ξ − 2η − φ4 + 2φ5 − 2φ6 − φ7 φ2 − φ1 − χ + ξ + 2η − 4φ3 + 7φ4 + 2φ5 − 2φ6 + 3φ7 φ1 + φ2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (32) and the energy density of the model is obtained as ρ = 1 4 �χ + ξ + 2η − 4φ3 + 7φ4 + 2φ5 − 2φ6 + 3φ7 φ1 + φ2 + χ + ξ − 2η − φ4 + 2φ5 − 2φ6 − φ7 φ2 − φ1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (33) The values of χ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' ξ and η are same for all the three cases whereas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' the values of φi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' for i=1 to 7 for the corresponding cases are clearly given in appendix section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' 4 Physical and geometrical properties The average Hubble parameter is H = α β coth(αt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (34) 8 From the figure of Hubble parameter, we trace that it decreases with the decrease of redshift i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' decreases as time increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' By choosing the values of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='21 and β = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='10 in the scale factor the Hubble parameter is obtained as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='07Gyrs−1 which is nearly equal to the present observational data [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' For this quantity the dimension is 1 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' By using this formula, we can also measure the age of the cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (ii) The volume of the model is given by Figure 1: Plot of Hubble parameter(H) versus redshift(z) V = a3 = (sinh(αt)) 3 β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (35) In figure 2, it is clear that the spatial volume increases with the decrease of redshift i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' it increases as the time increases and is finite at final epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (iii) The expansion scalar θ is Figure 2: Plot of volume(V ) versus redshift(z) θ = ui ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='i = 3H = 3α coth(αt) β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (36) From figure 3, it is observed that expansion scalar decreases with the decrease of redshift i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' it decreases as time increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Here we noticed that for t = 0 the expansion scalar is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (iv) We get the shear scalar as 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='25 Hubble parameter(H) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)30 25 Volume(V) 20 15 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)Figure 3: Plot of expansion scalar(θ) versus redshift(z) σ2 = 3α2(n − 1)2 coth2(αt) β2(n + 2)2 , (37) when t = 0, σ2 (shear scalar) tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (v) The mean anisotropy parameter Ah is obtained as Ah = 1 3 � 3 � i=1 �Hi − H H �2� (38) where i = 1, 2, 3 indicate the directional Hubble parameters for the coordinates of r, θ and ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The mean anisotropy parameter is defined on the basis of directional Hubble parameter and mean Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Ah = 2(n − 1)2 (n + 2)2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' n ̸= −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (39) The mean anisotropy parameter Ah is useful for checking if the model is anisotropic or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In the present model Ah = 0 for n = 1 and Ah ̸= 0 for n ̸= 1 that is the model is anisotropic for n ̸= 1 and isotropic for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In all the discussions and graphical representation of physical parameters we constraint the constants for case I as α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='21, β = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='10, n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='38, λ = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='02, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='03, case II as ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='001, τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='002 and case III as κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2, ι = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The values of parameters α, β, n, λ in cases II and III are same as that of Case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (iv) The deceleration parameter is q = −1 + d dt( 1 H ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (40) In this model by using hyperbolic function we obtained deceleration parameter as q = −1 + β(1 − tanh2(αt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (41) When t < 1 α tanh−1(1 − 1 β) 1 2, q has negative value which represents that the universe is acceler- ating whereas if t > 1 α tanh−1(1− 1 β) 1 2, q has positive value which represents that the universe is 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='8 expansion scalar(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)decelerating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The quantities such as q and H specifies the geometric properties of the cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' v)Throughout the plots uniform colouring is followed by giving the colours brown for pressure, navy blue for energy density, sky blue for EoS parameter, blue for SEC, green for NEC and red for DEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Figures 4,5 and 6 illustrate the variation of pressure against redshift in cases I, II and III respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The figures shows that in three cases pressure is negative and it is known that a negative pressure fluid is the correct mechanism which is capable of explaining cosmic acceleration within the standard cosmologies, despite the fact that in the latter it is necessary to bring the cosmological constant to get this exotic characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In pressure graphs increase with the decrease of redshift that it is increases as the time increases is perceived which repre- sents cosmic acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Figure 4: Pressure(p) in case I Figure 5: Pressure(p) in case II Figure 6: Pressure(p) in case III vi) Figures 7,8 and 9 shows the evolution of energy density for cases I, II and III respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In all the cases the density decreases with the decrease of redshift i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' decreases as time increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' vii) With great efforts the equation of state(EoS) parameter in cosmology of different dark Figure 7: Energy density(ρ) in case I Figure 8: Energy density(ρ) in case II Figure 9: Energy density(ρ) in case III energy models are examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The parameter relating to the equation of state is a dimen- sionless term that represents the matter state under some particular physical grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In the terminology of p and ρ the EoS can be interpret in the from of ω = p ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The EoS parameter 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='08 pressure(p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='1 pressure(p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='06 pressure(p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='18 energy density(p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='16 energy density(p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)is distinguished in three regions namely quintessence, phantom, and quintom according to its range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In quintessence region the EoS paramter lies in the range of −1 < ω < − 1 3, in phantom phase the EoS parameter is in the range of less than -1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' ω < −1) and in quintom ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Figures 10, 11 and 12 of EoS parameter are drawn against redshift and observe that decrease with the decrease of redshift that is decrease as time increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' From the graphs we noticed that our model lies in quintessence region in three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' According to Planck+nine years WMAP the current value of EoS parameter is approximately as ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='13+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='25 [58], and from SNe Ia data with galaxy clustering, CMBR anisotropy statistics the EoS parameter lies in the range −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='33 < ω < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='79, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='67 < ω < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='62 [59] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' From the figures of EoS parameters, it is seen that three cases are approximately coincide with observational data which is a good result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' viii)In modified theories of gravity, energy conditions [60–62] plays a crucial role in studying Figure 10: EoS parameter(ω) in case I Figure 11: EoS parameter(ω) in case II Figure 12: EoS parameter(ω) in case III the behaviour of spacelike and timelike geodesics and these conditions are came from Ray- chaudhuri equations [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Energy conditions can be defined in many ways, such as geometric way and physical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Moreover energy conditions are significant in the black hole physics, as they lay foundations of the singularity theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Another advantage of energy condition is that it allows basic tools to consider certain ideas about black holes and wormholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' There are four most commonly used fundamental energy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The general expressions for energy conditions in regard of pressure and energy density are given below: (i)SEC(Strong Energy condition): Gravity always has to be attractive, and in cosmology ρ + 3p ≥ 0, ρ + p ≥ 0 should be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (ii)DEC(Dominant Energy Condition): The energy density should always be positive when measured by any observer that is ρ ≥ 0, ρ ± p ≥ 0, must be obeyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (iii)WEC(Weak Energy Condition): The energy density must always be positive when mea- sured by any observer that is ρ ≥ 0, ρ + p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (iv)NEC(Null Energy Condition): NEC is expressed in the form of ρ+p ≥ 0 and it ensures the validity of second law of black hole thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 redshift(z)Where NEC, WEC, DEC and SEC represents null, weak, dominant and strong energy condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' According to present cosmological data to represent the universe with cosmic expansion the SEC of that model should be violated (ρ + 3p ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' For the obtained models the same scenario can be clearly observed from figures 13 to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' When compared to strong energy condi- tion null energy condition is more beneficial, as it can be used algebraically due to its weakest pointwise energy condition which results in the strongest theorems and all these energy condi- tions, are met by electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' From figures 16 to 18 it is clear that NEC(ρ + p ≥ 0) is satisfied in all the three cases for the obtained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' If NEC satisfies then the parameter EoS occurs in quintessence region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Also from figures 19 to 21 it is clear that DEC (ρ + p ≥ 0) is fulfilled in all the three cases for the obtained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Figure 13: SEC in case I Figure 14: SEC in case II Figure 15: SEC in case III Figure 16: NEC in case I Figure 17: NEC in case II Figure 18: NEC in case III Figure 19: DEC in case I Figure 20: DEC in case II Figure 21: DEC in case III 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='15 SEC(p+3p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='16 SEC(p+3p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='1 SEC(p+3p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='04 NEC(p+p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='05 NEC(p+p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='3 DEC(p-p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='35 DEC(p-p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='5 2 redshift(z)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='1 Stability analysis Perturbations are essential for simplify a complex mathematical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' There are several types of perturbations such as isotropic, anisotropic, homogeneous/inhomogeneous scalar, vec- tor and tensor perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The technique of perturbation is studied as a tool for finding approximate solution and comparing it to the obtained exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Some of the researchers who studied on stability analysis are [64–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Here the stability of solutions in terms of metric perturbation as following ai → aBi + δai = aBi(1 + δbi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (42) The perturbation of volume scale factor, directional Hubble factors and mean Hubble factors are V → VB + VB � i δbi, θi → θBi + � i δbi, θ → θB + 1 3 � i δbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (43) The following equations are satisfied by the metric perturbation δbi � i δ¨bi + 2 � i θBiδ˙bi = 0, (44) δ¨bi + ˙VB VB δ˙bi + � j δ˙bjθBi = 0, (45) � i δ˙bi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (46) From equations (44) - (46), we attain δ¨bi + ˙VB VB δ˙bi = 0, (47) where VB is the background spatial volume and for our case VB is VB = (sinh(αt)) 3 β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (48) From above two equations, δbi is procured as δbi = c1 − c �β � cosh2(αt) sech(αt) sinh β−3 β (αt)2F1 � 1 2, β−3 2β ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' 3(β−1) 2β ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' − sinh2(αt) � α(β − 3) � , (49) where c1 and c are integrating constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' consequently, the actual fluctuations δai = aBiδbi is δai = � c1−c �β � cosh2(αt) sech(αt) sinh β−3 β (αt)2F1 � 1 2, β−3 2β ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' 3(β−1) 2β ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' − sinh2(αt) � α(β − 3) �� (sinh(αt)) −3 β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' (50) 14 Figure 22: Plot of actual fluctuations(δai) versus redshift(z) Figure 22 shows the behaviour of actual fluctuations versus redshift and it is noticed that it is a decreasing function with the decrease of redshift that is decreases as time increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' It is clear that δai → 0 as z → −∞ and hence the background solution is shown to be stable against perturbation of gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' 5 Conclusions A cosmological model in f(R, T) theory with three cases namely power law, logarithmic and exponential curvature is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Hyperbolic scale factor is used to solve the field equations to get the solution in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The solutions of these field equations represent accelerating model of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The graph of all parameters are drawn against redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In graphs the negative region of z represents future epoch, positive region of z represents past and z = 0 indicates present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Obtained models are anisotropic and free from singularity all the way through the universe’s evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' By analyzing all the parameters the conclusions are as follows: From figures 1 and 3 and from the equations (34) and (36) it can be seen that Hubble parameter and expansion scalar decreases with the decrease of redshift, and also it is clear that the Hubble parameter and expansion scalar are close to zero when t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' From figure 2 it is clear that volume increases with the decrease of redshift which indicates volume of the expanding universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' From equation (37), it is noticed that the shear scalar is a function of time and tends to zero when t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' From equation (39), the anisotropic parameter is independent of time and Ah ̸= 0 for n ̸= 1, Ah = 0 for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' But in this paper due to power law n is different from one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Hence the models are anisotropic throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='02 0 1 0 1 2 3 redshift(z)• From the graphs of pressure and energy density of all the three cases, it is clear that the pressure and energy density are negative and positive respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Due to the negative pressure and positive energy density the universe is going through accelerating expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The behavior of EoS parameter against redshift is represented in plots 11 to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' From these graphs it is obvious that the model is in the quintessence region in three cases that is −1 < ω < − 1 3 which matches with present observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' In three cases, SEC is violated whereas NEC and DEC are fufilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' The violation of SEC leads to cosmic acceleration which is in good agreement with the expansion of the cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' As seen in the graph of stability analysis, the actual fluctuations begin with a small positive value and decreases to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' As a result, the background solution is stable when the gravitational field is perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' A detailed discussion is provided through the obtained models for describing cosmic accel- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Finally, through the detailed study of the models in three cases namely power law curvature f(R, T) = R+γR2 − µ4 R +λT, logarithmic curvaturef(R, T) = R+νln(τR)+λT and exponential curvature f(R, T) = R+κe−ιR+λT very good results which represents the universe accelerating expansion are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Moreover all the parameters discussed here matches with the recent observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' At last, without existence of any exotic fluid, the current uni- verse is accelerating is perceived in this paper which is a great outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' As a future work, this work can be extended to other anisotropic models and can study the similarities and differences between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' 6 Appendix The values of χ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' η are same for all the three cases and are given below χ = ϱ1 ϱ2 where ϱ1 = β2 sinh(αt)2(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2 sinh(αt) −6+(−2n−4)β β(n+2) −6(−9 cosh(αt)2 2 + β(n + 2))α2 ϱ2 = β2(n + 2)2 sinh(αt)2 ξ = ϱ3 ϱ2 16 where ϱ3 = −3 � (−3n2 − 3n − 3) cosh(αt)2 + β(n + 2)(n + 1) � α2 η = ϱ4 ϱ2 ϱ4 = β2 sinh(αt)2(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2 sinh(αt) −6+(−2n−4)β β(n+2) +18α2(n + 1 2) cosh(αt)2 For case(i) The values of φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ6 and φ7 are given below ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ1(t) = ϱ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ5(t) = 288(n + 2)2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6+(2n+4)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) β2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(n + 2)2β − 3n2 − 6n − 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2 cosh(αt)2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6+(2n+4)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+ cosh(αt)2β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='− β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(n + 2)2β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(π + 3λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ6(t) = ϱ7(t) + 4(cosh(αt) − 1)(n + 2)2β2(cosh(αt) + 1)ϱ8(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ7(t) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='µ4(n + 2)6β6 + 324α4(n + 2)2(n2 + 2n + 3)2β2 − 11664α6(n2 + 2n + 3)3γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)6 − 3(n + 2)2β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='µ4(n + 2)4β5 + 72α4(n2 + 2n + 3)(n + 2)2β2 + 108α4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(n2 + 2n + 3)2β − 3888α6(n2 + 2n + 3)2γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)4 + 3(n + 2)4β2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='µ4(n + 2)2β4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+ 12α4(n + 2)2β2 + 72α4(n2 + 2n + 3)β − 1296α6(n2 + 2n + 3)γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='− β3(n + 2)6(−432α6γ + µ4β3 + 36α4β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ8(t) = −6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(−3n2 − 6n − 9) cosh(αt)2 + (n + 2)2β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(−54α2(n2 + 2n + 3)γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+ β2(n + 2)2) cosh(αt)2 − β(n + 2)2(−18α2γ + β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) + (n + 2)2β2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(cosh(αt) − 1)(cosh(αt) + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='((−108α2(n2 + 2n + 3)γ + β2(n + 2)2) cosh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='− β(n + 2)2(−36α2γ + β)) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) − 4β2γ(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ2(t) = ϱ9(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ6(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ9(t) = λ18(n + 2)2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6+(2n+4)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) β2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(n + 2)2β − 3n2 − 6n − 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2 cosh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) + (n + 2)2β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6+(2n+4)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+ β sinh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ3(t) = ϱ10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ10(t) = −2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ11 − ϱ12 + β2(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ11(t) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='µ4(n + 2)6β6 − 2916α6γ(n2 + 2n + 3)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)6) − 3β(n + 2)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='µ4(n + 2)4β5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='− 972α6γ(n2 + n + 3)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)4 + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='µ4(n + 2)4β4 − 324α6γ(n2 + 2n + 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(n + 2)4 cosh(αt)2 − β3(n + 2)6(−108α6γ + β3µ4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ12(t) = 108β2(n + 2)2γ(cosh(αt) + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(−3n2 − 6n − 9) cosh(αt)2 + (n + 2)2β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(cosh(αt) − 1) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) + 36β4(n + 2)4γ(cosh(αt) + 1)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(−3n2 − 6n − 9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)2 + (n + 2)2β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2(cosh(αt) − 1)2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) − 4γβ6(cosh(αt) − 1)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(cosh(αt) + 1)3(n + 2)6� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(−3n2 − 6n − 9) cosh(αt)2 + (n + 2)2β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ13(t) = (ϱ7 − ϱ14) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2)(n + 2)2β2 sinh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ14(t) = −24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(n + 2)2β2 − 54α2γ(n2 + 2n + 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)2 − β(n + 2)2(−18α2γ + β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(n + 2)2α2(cosh(αt) − 1)β2(cosh(αt) + 1)((−3n2 − 6n − 9) cosh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+ (n + 2)2β) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) + 4(n + 2)4�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(n + 2)2β2 − 108α2γ(n2 + 2n + 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)2 − β(n + 2)2(−36α2γ + β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(cosh(αt) − 1)2β4(cosh(αt) + 1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) − 16γβ6(cosh(αt) − 1)3(cosh(αt) + 1)3(n + 2)6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ4(t) = −36α2ϱ15ϱ16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(ϱ7 − ϱ14)ϱ17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ15(t) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='µ4(n + 2)6β6 + 5832α6γ(n2 + 2n + 3)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)6 − 3β(n + 2)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='µ4(n + 2)4β5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+ 1944α6γ(n2 + 2n + 3)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)4 + 3β2(n + 2)4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='µ4(n + 2)2β4 + 648α6γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(n2 + 2n + 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)2 − β3(n + 2)6(216α6γ + β3µ4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) + ϱ25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ25(t) = 216(cosh(αt) + 1)β2(cosh(αt) − 1)(n + 2)2γα4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(−3n2 − 6n − 9) cosh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+ β(n + 2)2�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) − 72(cosh(αt) + 1)2β4(cosh(αt) − 1)2(n + 2)4γα4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(−3n2 − 6n − 9) cosh(αt)2 + β(n + 2)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) + 8γβ6(cosh(αt) − 1)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(cosh(αt) + 1)3(n + 2)6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ16(t) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(β(n + 2)2 − 3n2 − 6n − 9) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) − β(cosh(αt) − 1)(cosh(αt) + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(n + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ17(t) = (−3α2((−3n2 − 6n − 9) cosh(αt)2 + (n + 2)2β) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) + β2(cosh(αt) − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(cosh(αt) + 1)(n + 2)2)β(n + 2) sinh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ5(t) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='24ϱ15ϱ18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(ϱ7 − ϱ14)ϱ19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ18(t) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='((n + 2)2β − 3n2 − 6n − 9)α2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='−(cosh(αt) − 1)(6 cosh(αt)2 + β(n + 2))(cosh(αt) + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ19(t) = −3α2((−3n2 − 6n − 9) cosh(αt)2 + (n + 2)2β) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+β2(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ6(t) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ21ϱ22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ20(t) = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='((n + 2)2β − n2 − 6n − 9)α2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6+(2n+4)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='− ((n + 2)2) − 3n2 − 6n − 9) cosh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) + β sinh(αt)2(n + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='18+(4n+8)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β6 cosh(αt)2(n + 2)6α2µ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ21(t) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2β(n + 2)2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6+(2n+4β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='− ((n + 2)2β − 3n2 − 6n − 9) cosh(αt)2α2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+β2 sinh(αt)2(n + 2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ22(t) = ϱ23 + ϱ24(n + 2)2β2(cosh(αt) − 1)(cosh(αt) + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ23(t) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ26 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='8((n + 2)2β(µ4(n + 2)4β5 + 72α4(n2 + 2n + 3)(n + 2)2β2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+ 108α4(n2 + 2n + 3)2β − 3888α6(n2 + 2n + 3)2γ) cosh(αt)4) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='8(n + 2)4β2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(µ4(n + 2)2β4 + 12α4(n + 2)2β2 + 72α4(n2 + 2n + 3)β − 1296α6(n2 + 2n + 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='γ) cosh(αt)2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='24β3(n + 2)6(432α6γ + µ4β3 + 36α4β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ26(t) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�−µ4(n + 2)6β6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='− 27α4(n + 2)2(n2 + 2n + 3)2β2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+ 486α6(n2 + 2n + 3)3γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ24(t) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='((−3n2 − 6n − 9) cosh(αt)2 + (n + 2)2β)α2((−54α2(n2 + 2n + 3)γ + β2(n + 2)2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)2 − β(n + 2)2(−18α2γ + β)) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2(n + 2)2β2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(−β2(n + 2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+ 27α2(n2 + 2n + 3)γ) cosh(αt)2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='4β(n + 2)2(−36α2γ + β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) + β2γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(cosh(αt) − 1)(cosh(αt) + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ7(t) = −36nα2ϱ15ϱ16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ϱ17(ϱ7 − ϱ14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='For case(ii) The values of φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ6 and φ7 are given below ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ1(t) = 48δ3(π + 3λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ1 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ν(n + 2)2β2 − 18α2(n2 + 2n + 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)2 − β(n + 2)2(−6α2 + βν) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) − 2β2(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ3 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(−3n2 − 6n − 9) cosh(αt)2 + β(n + 2)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='−β2 sinh(αt)2(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ2(t) = −δ2λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ2 = −3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(−3n2 − 6n − 9) cosh(αt)2 + β(n + 2)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+β2(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ3(t) = −3νδ3δ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ4 = ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β2(n + 2)2 sinh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+ ln(2) + ln(τ) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ9 = −β2(cosh(αt) − 1)2(cosh(αt) + 1)2(n + 2)2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='−6+(−2n−4)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+3α2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(−3n2 − 6n − 9) cosh(αt)2 + β(n + 2)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ4(t) = δ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ2δ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ5 = 18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='− β(cosh(αt) − 1)(cosh(αt) + 1)(n + 2) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) + (β(n + 2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='−3n2 − 6n − 9) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2)α2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(n + 2)νβ cosh(αt)2α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ5(t) = δ6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ2δ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ6 = −12(n + 2)2νβ2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ10 + (β(n + 2)2 − 3n2 − 6n − 9)(cosh(αt)2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2)α2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ10 = −(cosh(αt) − 1)(cosh(αt) + 1)(6 cosh(αt)2 + β(n + 2)) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ6(t) = δ7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ8δ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ7 = 4(n + 2)2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2)νβ2 cosh(αt)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(β(n + 2)2 − 3n2 − 6n − 9)α2] sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='−β(cosh(αt) − 1)(cosh(αt) + 1)(n + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ8 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='− (β(n + 2)2 − 3n2 − 6n − 9) cosh(αt)2α2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) + δ11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ11 = (n + 2)2β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6+(2n+4)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+ β sinh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ7(t) = nδ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='δ2δ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='For case(iii) The values of φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' φ6 and φ7 are given below ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ1(t) = −16π − 3λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2ζ1 − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ζ1 = κιe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='−2ι ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ζ8−3α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(−3n2−6n−9) cosh(αt)2+β(n+2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β2(n+2)2 sinh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ζ8 = β2(cosh(αt) − 1)2(cosh(αt) + 1)2(n + 2)2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='−6+(−2n−4)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ2(t) = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2ζ1 − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ3(t) = ζ2ζ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ζ2 = e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β2 sinh(αt)2(cosh(αt)4+1)(n+2)2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='−6+(−4n−8)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+9α2 cosh(αt)2(n2+2n+3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ι ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β2(n+2)2 sinh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ζ3 = β2(n + 2)2 sinh(αt)4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ικe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2ι ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β2(cosh(αt)4+1)(n+2)2 sinh(αt) −6+(−2n−4)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+9α2 cosh(αt)2(n2+2n+3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β2(n+2)2 sinh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='−e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='4ι ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) −6+(−2n−4)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)2β+ 3α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β sinh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ζ4 = κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β2ι(cosh(αt) − 1)(cosh(αt) + 1)(n + 2)2 sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�−(n + 2)2β2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='+9α2ι(n2 + 2n + 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)2 − 3(n + 2)2β(ια2 − β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ4(t) = −36α2ζ5ζ2ι2κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n + 2)ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ζ5 = β(cosh(αt) − 1)(cosh(αt) + 1)(n + 2) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) + α2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(n + 2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β − 3n2 − 6n − 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ5(t) = −24ι2ζ2α2κζ6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ζ6 = −ζ8 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='cosh(αt)2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='((n + 2)2β − 3n2 − 6n − 9)α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ζ8 = (cosh(αt) − 1)(6 cosh(αt)2 + β(n + 2))(cosh(αt) + 1) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ6(t) = −144α2ι2 cosh(αt)2ζ1ζ7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(n + 2)4β4(ζ1 − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='φ7(t) = −36nα2ζ5ζ2ι2κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n + 2)ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='ζ7 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='α2((n + 2)2β − 3n2 − 6n − 9) sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) − β(cosh(αt) − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='(cosh(αt) + 1)(n + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='sinh(αt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='−12+(−6n−12)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='β(n+2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='References ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content='[1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfOPuo/content/2301.01163v1.pdf'} +page_content=' Riess et al.' metadata={'source': 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b/rdA0T4oBgHgl3EQfK_86/content/tmp_files/2301.02111v1.pdf.txt @@ -0,0 +1,988 @@ +Neural Codec Language Models are +Zero-Shot Text to Speech Synthesizers +Chengyi Wang∗ Sanyuan Chen∗ Yu Wu∗ Ziqiang Zhang Long Zhou Shujie Liu +Zhuo Chen Yanqing Liu Huaming Wang Jinyu Li Lei He Sheng Zhao Furu Wei +Microsoft +https://github.com/microsoft/unilm +Abstract +We introduce a language modeling approach for text to speech synthesis (TTS). +Specifically, we train a neural codec language model (called VALL-E) using +discrete codes derived from an off-the-shelf neural audio codec model, and re- +gard TTS as a conditional language modeling task rather than continuous signal +regression as in previous work. During the pre-training stage, we scale up the TTS +training data to 60K hours of English speech which is hundreds of times larger than +existing systems. VALL-E emerges in-context learning capabilities and can be +used to synthesize high-quality personalized speech with only a 3-second enrolled +recording of an unseen speaker as an acoustic prompt. Experiment results show +that VALL-E significantly outperforms the state-of-the-art zero-shot TTS system +in terms of speech naturalness and speaker similarity. In addition, we find VALL-E +could preserve the speaker’s emotion and acoustic environment of the acoustic +prompt in synthesis. See https://aka.ms/valle for demos of our work. +Figure 1: The overview of VALL-E. Unlike the previous pipeline (e.g., phoneme → mel-spectrogram +→ waveform), the pipeline of VALL-E is phoneme → discrete code → waveform. VALL-E +generates the discrete audio codec codes based on phoneme and acoustic code prompts, corresponding +to the target content and the speaker’s voice. VALL-E directly enables various speech synthesis +applications, such as zero-shot TTS, speech editing, and content creation combined with other +generative AI models like GPT-3 [Brown et al., 2020]. +∗These authors contributed equally to this work. Correspondence: {yuwu1,shujliu,fuwei}@microsoft.com +arXiv:2301.02111v1 [cs.CL] 5 Jan 2023 + +Personalized +Speech +VALL-E +Audio CodecDecoder +Neural Codec Language Modeling +PhonemeConversion +Audio Codec Encodel +Text +Acoustic +Prompt +Prompt +Text for synthesis +3-second enrolled recording1 +Introduction +The last decade has yielded dramatic breakthroughs in speech synthesis through the development of +neural networks and end-to-end modeling. Currently, cascaded text to speech (TTS) systems [Shen +et al., 2018, Ren et al., 2019, Li et al., 2019] usually leverage a pipeline with an acoustic model and a +vocoder using mel spectrograms as the intermediate representations. While advanced TTS systems +can synthesize high-quality speech from single or multiple speakers [Liu et al., 2022, Kim et al., +2021], it still requires high-quality clean data from the recording studio. Large-scale data crawled +from the Internet cannot meet the requirement, and always lead to performance degradation. Because +the training data is relatively small, current TTS systems still suffer from poor generalization. Speaker +similarity and speech naturalness decline dramatically for unseen speakers in the zero-shot scenario. +To tackle the zero-shot TTS problem, existing work leverages speaker adaptation [Chen et al., 2019, +Wang et al., 2020] and speaker encoding [Arik et al., 2018, Casanova et al., 2022b] methods, requiring +additional fine-tuning, complex pre-designed features, or heavy structure engineering. +Instead of designing a complex and specific network for this problem, the ultimate solution is to +train a model with large and diverse data as much as possible, motivated by success in the field of +text synthesis [Brown et al., 2020, Chowdhery et al., 2022]. Recent years have witnessed notable +performance improvement for data increase in the text language model, from 16GB of uncompressed +text [Devlin et al., 2019], to 160GB [Liu et al., 2019], to 570GB [Brown et al., 2020], and finally, +around 1TB [Chowdhery et al., 2022]. Transferring this success to the field of speech synthesis, we +introduce VALL-E, the first language model based TTS framework leveraging the large, diverse, and +multi-speaker speech data. As shown in Figure 1, to synthesize personalized speech (e.g., zero-shot +TTS), VALL-E generates the corresponding acoustic tokens conditioned on the acoustic tokens of +the 3-second enrolled recording and the phoneme prompt, which constrain the speaker and content +information respectively. Finally, the generated acoustic tokens are used to synthesize the final +waveform with the corresponding neural codec decoder [Défossez et al., 2022]. The discrete acoustic +tokens derived from an audio codec model enable us to treat TTS as conditional codec language +modeling, and advanced prompting-based large-model techniques (as in GPTs [Brown et al., 2020]) +can be leveraged for the TTS tasks. The acoustic tokens also allow us to generate diverse synthesized +results in TTS by using different sampling strategies during inference. +We train VALL-E with LibriLight [Kahn et al., 2020], a corpus consisting of 60K hours of English +speech with over 7000 unique speakers. The original data is audio-only, so we employ a speech +recognition model to generate the transcriptions. Compared to previous TTS training datasets, such +as LibriTTS [Zen et al., 2019], our data contain more noisy speech and inaccurate transcriptions but +provide diverse speakers and prosodies. We believe the proposed approach is robust to the noise and +generalize well by leveraging large data. It is worth noting that existing TTS systems are always +trained with dozens of hours of single-speaker data or hundreds of hours of multi-speaker data, which +is over hundreds of times smaller than VALL-E. Table 1 summarizes the innovation of VALL- +E, a language model approach for TTS, using audio codec codes as intermediate representations, +leveraging large and diverse data, leading to strong in-context learning capabilities. +Table 1: A comparison between VALL-E and current cascaded TTS systems. +Current Systems +VALL-E +Intermediate representation +mel spectrogram +audio codec code +Objective function +continuous signal regression +language model +Training data +≤ 600 hours +60K hours +In-context learning + + +We evaluate VALL-E on LibriSpeech [Panayotov et al., 2015] and VCTK [Veaux et al., 2016] +datasets, where all test speakers are unseen in the training corpus. VALL-E significantly outperforms +the state-of-the-art zero-shot TTS system [Casanova et al., 2022b] in terms of speech naturalness and +speaker similarity, with +0.12 comparative mean option score (CMOS) and +0.93 similarity mean +option score (SMOS) improvement on LibriSpeech. VALL-E also beats the baseline on VCTK with ++0.11 SMOS and +0.23 CMOS improvements. It even achieves a +0.04 CMOS score against ground +truth, showing the synthesized speech of unseen speakers is as natural as human recordings on VCTK. +Moreover, the qualitative analysis shows that VALL-E is able to synthesize diverse outputs with the +2 + +same text and target speaker, which could benefit pseudo-data creation for the speech recognition task. +We also find that VALL-E could keep the acoustic environment (e.g., reverberation) and emotion +(e.g. anger) of the acoustic prompt. +In summary, we make the following contributions. +• We propose VALL-E, the first TTS framework with strong in-context learning capabilities as +GPT-3, which treats TTS as a language model task with audio codec codes as an intermediate +representation to replace the traditional mel spectrogram. It has in-context learning capability +and enables prompt-based approaches for zero-shot TTS, which does not require additional +structure engineering, pre-designed acoustic features, and fine-tuning as in previous work. +• We build a generalized TTS system in the speaker dimension by leveraging a huge amount +of semi-supervised data, suggesting that simple scaling up semi-supervised data has been +underestimated for TTS. +• VALL-E is able to provide diverse outputs with the same input text and keep the acoustic +environment and speaker’s emotion of the acoustic prompt. +• We verify that VALL-E synthesizes natural speech with high speaker similarity by prompt- +ing in the zero-shot scenario. Evaluation results show that VALL-E significantly outper- +forms the state-of-the-art zero-shot TTS system on LibriSpeech and VCTK. +We encourage the reader to listen to our samples on the demo page https://aka.ms/valle. +2 +Related Work +Zero-Shot TTS: Current TTS methods can be categorized into cascaded and end-to-end methods. +Cascaded TTS systems [Shen et al., 2018, Ren et al., 2019, Li et al., 2019] usually leverage a pipeline +with an acoustic model and a vocoder using mel spectrograms as the intermediate representations. To +tackle the drawbacks of the vocoder, end-to-end TTS models [Kim et al., 2021, Liu et al., 2022] are +proposed to jointly optimize the acoustic model and vocoder. In real scenarios, it is highly desirable +to customize a TTS system to an arbitrary voice with rare enrolled recordings. Therefore, there is +growing interest in the zero-shot multi-speaker TTS techniques, and most of work is done in the +context of cascaded TTS systems. As the pioneers, Arik et al. [2018] proposes speaker adaptation +and speaker encoding approaches. In the line of speaker adaptation, the following work [Chen +et al., 2019, Wang et al., 2020, Chen et al., 2021] tries to improve the adaptation efficiency with less +target speaker data and speaker-specific parameters. Huang et al. [2022] applies meta-learning on +speaker adaptation, which only requires 5-shot to build a well-performed system. In parallel, speaker +encoding-based methods achieved great progress in recent years. A speaker encoding based system +contains a speaker encoder and a TTS component, where the speaker encoder could be pre-trained +on the speaker verification task [Jia et al., 2018]. In Jia et al. [2018] and Arik et al. [2018], the +experiments show that the model is able to generate high-quality outputs with 3 seconds enrolled +recordings for in-domain speakers. To improve the quality of unseen speakers, advanced speaker +embedding models [Cai et al., 2018] can be employed, but it is still undesirable according to Tan +et al. [2021]. Another way is to design advanced but complex speaker encoder [Wu et al., 2022]. +Diffusion model based TTS [Popov et al., 2021, Kim et al., 2022] is also extended to zero-shot TTS +[Kang et al., 2022] and achieved good results. Compared to previous work [Ren et al., 2019, Du et al., +2022], our work follows the line of cascaded TTS but first uses audio codec code as intermediate +representations. It is the first one that has strong in-context learning capabilities as GPT-3, which +does not require fine-tuning, pre-designed features, or a complex speaker encoder. +Spoken generative pre-trained models: Self-supervised learning is widely investigated in the field +of speech understanding [Baevski et al., 2020b, Hsu et al., 2021, Chen et al., 2022] and speech-to- +speech generation [Lakhotia et al., 2021, Borsos et al., 2022]. In the context of speech-to-speech +generation, a hot topic is how to synthesize speech in a textless setting. GSLM [Lakhotia et al., +2021] proposes to synthesize speech based on HuBERT codes [Hsu et al., 2021], and Polyak et al. +[2021] improves the performance by combining HuBERT codes with codes of VQVAE and a speaker +encoder. AudioLM [Borsos et al., 2022] follows a similar way but use audio codecs [Zeghidour et al., +2022] to synthesize speech, together with semantic codes. It should be noted that AudioLM is able to +synthesize speech based on audio codecs without training an additional vocoder such as HifiGAN +[Kong et al., 2020]. AudioLM is a speech-to-speech model, whereas VALL-E is a TTS model, so +3 + +Figure 2: The neural audio codec model revisit. Because RVQ is employed, the first quantizer plays +the most important role in reconstruction, and the impact from others gradually decreases. +we can explicitly control the content in speech synthesis. Another direction is to apply pre-training +to the neural TTS. Chung et al. [2018] pre-trains speech decoder in TTS through autoregressive +mel-spectrogram prediction. In Ao et al. [2022], the authors propose a unified-modal encoder-decoder +framework SpeechT5, which can leverage unlabeled speech and text data to pre-train all components +of TTS model. Tjandra et al. [2019] quantizes unlabeled speech into discrete tokens by a VQVAE +model [van den Oord et al., 2017], and train a model with the token-to-speech sequence. They +demonstrate that the pre-trained model only requires a small amount of real data for fine-tuning. Bai +et al. [2022] proposes mask and reconstruction on mel spectrogram and showing better performance +on speech editing and synthesis. Previous TTS pre-training work leverages less than 1K hours of +data, whereas VALL-E is pre-trained with 60K hours of data. Furthermore, VALL-E is the first to +use audio codec codes as intermediate representations, and emerge in-context learning capability in +zero-shot TTS. +3 +Background: Speech Quantization +Since audio is typically stored as a sequence of 16-bit integer values, a generative model is required +to output 216 = 65, 536 probabilities per timestep to synthesize the raw audio. In addition, the audio +sample rate exceeding ten thousand leads to an extraordinarily long sequence length, making it more +intractable for raw audio synthesis. To this end, speech quantization is required to compress integer +values and sequence length. µ-law transformation can quantize each timestep to 256 values and +reconstruct high-quality raw audio. It is widely used in speech generative models, such as WaveNet +[van den Oord et al., 2016], but the inference speed is still slow since the sequence length is not +reduced. Recently, vector quantization is widely applied in self-supervised speech models for feature +extraction, such as vq-wav2vec [Baevski et al., 2020a] and HuBERT [Hsu et al., 2021]. The following +work [Lakhotia et al., 2021, Du et al., 2022] shows the codes from self-supervised models can also +reconstruct content, and the inference speed is faster than WaveNet. However, the speaker identity has +been discarded and the reconstruction quality is low [Borsos et al., 2022]. AudioLM [Borsos et al., +2022] trains speech-to-speech language models on both k-means tokens from a self-supervised model +and acoustic tokens from a neural codec model, leading to high-quality speech-to-speech generation. +In this paper, we follow AudioLM [Borsos et al., 2022] to leverage neural codec models to represent +speech in discrete tokens. To compress audio for network transmission, codec models are able to +encode waveform into discrete acoustic codes and reconstruct high-quality waveform even if the +speaker is unseen in training. Compared to traditional audio codec approaches, the neural-based +codec is significantly better at low bitrates, and we believe the quantized tokens contain sufficient +information about the speaker and recording conditions. Compared to other quantization methods, +the audio codec shows the following advantages: 1) It contains abundant speaker information and +acoustic information, which could maintain speaker identity in reconstruction compared to HuBERT +codes [Hsu et al., 2021]. 2) There is an off-the-shelf codec decoder to convert discrete tokens into a +waveform, without the additional efforts on vocoder training like VQ-based methods that operated on +spectrum [Du et al., 2022]. 3) It could reduce the length of time steps for efficiency to address the +problem in µ-law transformation [van den Oord et al., 2016]. +4 + +12438...59 +712138..67 +91652..84 +Quantized Tokens +Encoder +stage8 +91652 +Decoder +84 +VQ +8 +stage 2 +21 +38 +67 +stage 1 +12 +43 +59 +residual 1 +residual 2 +residual 7We adopt a pre-trained neural audio codec model, EnCodec [Défossez et al., 2022], as our tokenizer. +EnCodec is a convolutional encoder-decoder model, whose input and output are both 24 kHz audio +across variable bitrates. The encoder produces embeddings at 75 Hz for input waveforms at 24 kHz, +which is a 320-fold reduction in the sampling rate. Each embedding is modeled by a residual vector +quantization (RVQ), in which we choose eight hierarchy quantizers with 1024 entries each as shown +in Figure 2. This configuration corresponds to EnCodec at 6K bitrates for 24 kHz audio reconstruction. +In this setting, given a 10-second waveform, the discrete representation is a matrix with 750 × 8 +entries, where 750 = 24,000×10 +320 +is the downsampled time step and 8 is the number of quantizers. It +is fine to choose other bitrate settings. A larger bitrate corresponds to more quantizers and better +reconstruction quality. For example, if we choose EnCodecc at 12K bitrates, there are 16 quantizers +are needed and the 10-second waveform corresponds to a matrix with 750 × 16 entries. With the +discrete codes from all quantizers, the convolutional decoder of EnCodec generates real-valued +embeddings and reconstructs the waveform at 24 kHz. +4 +VALL-E +4.1 +Problem Formulation: Regarding TTS as Conditional Codec Language Modeling +Given a dataset D = {xi, yi}, where y is an audio sample and x = {x0, x1, . . . , xL} is its corre- +sponding phoneme transcription, we use a pre-trained neural codec model to encode each audio +sample into discrete acoustic codes, denoted as Encodec(y) = CT ×8, where C represents the +two-dimensional acoustic code matrix, and T is the downsampled utterance length. The row vec- +tor of each acoustic code matrix ct,: represents the eight codes for frame t and the column vector +of each acoustic code matrix c:,j represents the code sequence from the j-th codebook, where +j ∈ {1, . . . , 8}. After quantization, the neural codec decoder is able to reconstruct the waveform, +denoted as Decodec(C) ≈ ˆy. +Zero-shot TTS requires the model to synthesize high-quality speech for unseen speakers. In this +work, we regard zero-shot TTS as a conditional codec language modeling task. We train a neural +language model to generate an acoustic code matrix C conditioned on a phoneme sequence x and +an acoustic prompt matrix ˜CT ′×8 with the optimization objective of max p(C|x, ˜C). Here, ˜C is +obtained by the same neural codec with an enrolled recording as the input. We expect the neural +language model learns to extract the content and speaker information from the phoneme sequence +and the acoustic prompt, respectively. During inference, given a phoneme sequence and a 3-second +enrolled recording of the unseen speaker, the acoustic code matrix with corresponding content and +speaker’s voice is firstly estimated by the trained language model. Then the neural codec decoder +synthesizes the high-quality speech. +4.2 +Training: Conditional Codec Language Modeling +The neural speech codec model allows us to operate on discrete audio representations. Due to residual +quantization in the neural codec model, the tokens have a hierarchical structure: tokens from previous +quantizers recover acoustic properties like speaker identity, while the consecutive quantizers learn +fine acoustic details. Each quantizer is trained to model the residual from the previous quantizers. +Motivated by this, we design two conditional language models in a hierarchical manner. +For the discrete tokens from the first quantizer c:,1, we train an autoregressive (AR) decoder-only +language model. It is conditioned on the phoneme sequence x and the acoustic prompt ˜C:,1, +formulated as +p(c:,1|x, ˜C:,1; θAR) = +T +� +t=0 +p(ct,1|c +𝒄𝟏,𝟏 +𝒄𝟐,𝟏 +𝒄𝟎,𝒋 +𝒄𝟏,𝒋 +𝒄𝑻,𝒋 +… +𝒄𝟎 +𝒄𝟏 +෩𝑪 +𝒙 +෩𝑪 +𝒙 +𝒄𝟎 +𝒄𝟏 +Allow attend +Disallow attend +෩𝑪 +𝒙 +෩𝑪 +𝒙 +𝒄𝟎 +𝒋−𝟏 +𝒄𝟏 +𝒋−𝟏 +𝒄𝟎 +𝒋−𝟏 +𝒄𝟏 +𝒋−𝟏 +AR: 𝑐𝑖 only attends to left +NAR: attend to all tokens +Text +EnCodec +G2P +𝒙 +Text +EnCodec +G2P +𝒙 +𝒄𝟎,𝟏 +𝒄𝟎,𝟏:𝒋−𝟏 +𝒄𝟏,𝟏:𝒋−𝟏 +𝒄𝑻,𝟏:𝒋−𝟏 +෩𝑪 +෤𝒄𝟎,𝟏 +෤𝒄𝑻′,𝟏 +… +𝒄𝟎,𝟏 +෤𝒄𝟎,𝟏 +NAR ID 𝒋 +Conditional Codec Language Modeling +Figure 3: The structure of the conditional codec language modeling, which is built in a hierarchical +manner. In practice, the NAR decoder will be called seven times to generate codes in seven quantizers. +conditioned on the phoneme sequence x, the acoustic prompt ˜C and the predicted acoustic tokens +belong to the previous codebooks C:, tokens are appended after each of them. We compute sinuous position embedding +separately for prompt and input tokens. For the causal transformer model, each token ct,1 can attend +to (x, c≤t,1) as illustrated in the left part of Figure 3. The model is optimized to maximize the +probability of the next token in the first codebook. We share the parameters of the output projection +layer with the parameters of the acoustic embedding Wa. +In the AR model, we do not explicitly extract an audio clip as the prompt in training. The training +process is pure casual language model training. In this way, any prefix sequence c